Jung and Jungians
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Analytical Odysseys.
Предполагаемое знание феноменальных определений диссоциации, узнаваемая видимость исследований детерминированности сознания,
å utpeke øyeblikk sannheter inneholder viktige punkter for å forstå smarte brennende grunner tilfredshet innbefatter undersøkelser vitenskap,
disaccordi giustificati distinguendo contraddizioni osservazioni vuote scetticismo oggetti interi movimenti interi sistemi esistenza astratta,
ٹھوس جانکاری فوری حدود کی باتوں کا حواس خالص مختلف طریقوں سے پیچیدہ روابط اہم فصلوں کی عکاسی کے معاملات,
podstawowe uniwersalne prawdy obojętne relacje zachowujące zrozumiałe słowa uwierzytelnienia świadomość twierdzenia filozoficzne,
experiencias sensoriales que afirman sabidurías esferas profundas que devuelven sentidos dialécticos punteros del supuesto aprendizaje de la pluralidad contraria,
媒体の無関心な認識を考慮する極度の内なることは法を変えること規則を否定すること側面楽しむこと基礎変わらないこと規則否定規則依存性側面楽しさ基盤変わらない移動活動の本質的な機能動物の惨めさ.
Steve.D.Hammond.
As I mentioned before, everywhere you look you will see a Bride. They are just like icons placed in the correct place and in the correct moment. This Bride was kind of risking her life as she was on top of a house with no protection at all. You have to do what it takes to take the shot you want isn't it? I hope you like my take.
Thank you all for your appreciation.
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Even when there is not enough light, there is always some shot to take. The married couple didn't want to finish taking pictures and we took these ones after the party. The are so in love!.
Thank you all for your appreciation.
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Mark this day as it's the day I managed to get 10 Million views on my flickr account. It's been really busy this year and I managed to get all this views in just under a year. Thanks all for your appreciation and for spending time looking at my pictures and liking them and commenting on them.
This is certainly a great milestone.
Thanks for all the views and the likes and keep spreading the love for photography.
I've created my own tool to monitor all the likes and views of my account. The tool is still under construction but you can follow progress here:
You can find my solution on Github.
If you want to raise any issue on the app, you can do it here:
github.com/JordiCorbilla/FlickrPhotoStats/issues
Description of the application here
Thank you all for your appreciation.
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© 2015 Jordi Corbilla - All Rights Reserved.
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without permission.
I took a break from my large scene to smash together this behemoth. I wanted a machine that dwarfed the minifigures beside it but was also operational and could be built with real bricks. The most basic chasis is based on the Land Bounty Ninjago set -- the steering is actually a straigh rip. All in all, this was an immensly satisfying vehicle to build, and an even more satisfying vehicle to animate.
I used Mecabricks' new animation system to rig the main features. This made it much easier to position for rendering and allowed me to make a full animation showcase. You can find the model and animation on Mecabricks. Feel free to look through the node system I used but be warned: There is no IK system so far so the 4-bar linkage for the extension arms is a super messy pseudo-analytical jumble. If you do this yourself, pay special attention to when the rotation coordinate system flips.
And of course, while I was building this, I came up with an idea for another vehicle that would make this look like a hotwheels car...
Look at this, a second post on here within two weeks of my last! That hasn't happened in a while, but I'm hoping to change that and be a little more active on here as I still love LEGO. I just haven't had much time for it recently, but I'll be trying to do my best to build more. Anyway, this is an older build, which was actually built for and displayed at Bricks Cascade 2018. I'm not happy with the picture, but they all turned out fairly meh and this has long since been scrapped. Hope you guys enjoy it despite that!
Welcome to the cabbage patch, kid! Your future: the Fourth Industrial Revolution, the Forth Industrial Evolution. The Nano World Order! The Graphene Matrix! It’s a new era of profound degeneration, where organisms become algorithms. Human hacks: biochemical processes, electronic signals, store, analyze, no escape. Reengineering human life—666 half human half Beast.
Psychotronics: hacking the mind, brain, and consciousness. They will get inside your head: seeing through your eyes, hearing through your ears, reading your thoughts, inducing thoughts. They will hack you through the Mark of the Beast. They will plug you into the Hive Mind Beast System. Then you will be a brainless zombie of the Antichrist System. Your mind will be controlled with different frequencies and wave forms—controlling your psyche, modifying your consciousness. The New World Order Military Industrial Intelligence Complex, researching and developing highly sophisticated state of the art technology to harness the computing power of the human brain.
Bioenergetics, bio-photonics, biophysics, psionics, psycho-energetics, psychoneuroimmunology, quantum biology, radionics, scalar electromagnetic, bioelectromagnetism, biophotons, biopotentials, morphogenetic fields, non-hertzian waves, quantum fields, scalar waves, zero-point energy, 666 hacks, the path to transhumanism.
AI-powered analytic disclaimer:
“Public digital conversations provide unique insights on social trends shaping society’s opinion. We analyze key influencers and evolving themes as it is critical to understand how controversies and the public reaction unfold in real-time. We quickly identify any coordinated or unauthentic behavior aimed at fueling social unrest, polarization or pollution of the public debate.”
Now a commercial sponsored by the World Economic Forum:
Klaus Schwab- “Pay insufficient attention to the frightening scenario of a comprehensive cyber attack which will bring to a complete halt to the power supply, transportation, hospital services, our society as a whole. The covid-19 crisis would be seen in this respect as a small disturbance in comparison to a major cyber attack.”
Matthew 24:21 “For at that time there will be a great tribulation (pressure, distress, oppression), such as has not occurred since the beginning of the world until now, nor ever will [again].”
Take the jab, insert the microchip, become a better you…bahahahaha!!! I mean: it’s for your health, for your safety, for your own good, and for the good of society! #666
The Jack Welch College of Business and the Office of Alumni Engagement presented “Careers in Analytics” on April 10, 2019, at the Martire Forum. The alumni panel featured Justin Baigert ’05, vice president, Data & Analytics at GE, Joseph Lucibello ’11, senior manager, data scientist at WWE and Suzanne May ’13, research manager at Purchased. The moderator was Khawaja Mamun, associate professor of economics. Photo by Mark F. Conrad
The Jack Welch College of Business and the Office of Alumni Engagement presented “Careers in Analytics” on April 10, 2019, at the Martire Forum. The alumni panel featured Justin Baigert ’05, vice president, Data & Analytics at GE, Joseph Lucibello ’11, senior manager, data scientist at WWE and Suzanne May ’13, research manager at Purchased. The moderator was Khawaja Mamun, associate professor of economics. Photo by Mark F. Conrad
In this article im gonna share with you a deep understanding of the Instagram analytics.
Instagram is the most popular social networking platform these days.
If you want to market your products and services then it’s the best ever platform for you.
All you need to make a free account and start promoting your products and services.
www.coremafia.com/instagram-analytics-to-grow-engagement-...
“The privilege of a lifetime is to become who you truly are.”
Words by Dr. C. G. Jung - 12.58
Dr. Carl Jung was born today - July 26, 1875
Remembering a truly visionary man.
'Thank You, Sarah for reminding me....
for Flickriver - Sophie Shapiro
Different forms of fluctuations of the terrestrial gravity field are observed by gravity experiments. For example, atmospheric pressure fluctuations generate a gravity-noise foreground in measurements with super-conducting gravimeters. Gravity changes caused by high-magnitude earthquakes have been detected with the satellite gravity experiment GRACE, and we expect high-frequency terrestrial gravity fluctuations produced by ambient seismic fields to limit the sensitivity of ground-based gravitational-wave (GW) detectors. Accordingly, terrestrial gravity fluctuations are considered noise and signal depending on the experiment. Here, we will focus on ground-based gravimetry. This field is rapidly progressing through the development of GW detectors. The technology is pushed to its current limits in the advanced generation of the LIGO and Virgo detectors, targeting gravity strain sensitivities better than 10−23 Hz−1/2 above a few tens of a Hz. Alternative designs for GW detectors evolving from traditional gravity gradiometers such as torsion bars, atom interferometers, and superconducting gradiometers are currently being developed to extend the detection band to frequencies below 1 Hz. The goal of this article is to provide the analytical framework to describe terrestrial gravity perturbations in these experiments. Models of terrestrial gravity perturbations related to seismic fields, atmospheric disturbances, and vibrating, rotating or moving objects, are derived and analyzed. The models are then used to evaluate passive and active gravity noise mitigation strategies in GW detectors, or alternatively, to describe their potential use in geophysics. The article reviews the current state of the field, and also presents new analyses especially with respect to the impact of seismic scattering on gravity perturbations, active gravity noise cancellation, and time-domain models of gravity perturbations from atmospheric and seismic point sources. Our understanding of terrestrial gravity fluctuations will have great impact on the future development of GW detectors and high-precision gravimetry in general, and many open questions need to be answered still as emphasized in this article.
Keywords: Terrestrial gravity, Newtonian noise, Wiener filter, Mitigation
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Introduction
In the coming years, we will see a transition in the field of high-precision gravimetry from observations of slow lasting changes of the gravity field to the experimental study of fast gravity fluctuations. The latter will be realized by the advanced generation of the US-based LIGO [1] and Europe-based Virgo [7] gravitational-wave (GW) detectors. Their goal is to directly observe for the first time GWs that are produced by astrophysical sources such as inspiraling and merging neutron-star or black-hole binaries. Feasibility of the laser-interferometric detector concept has been demonstrated successfully with the first generation of detectors, which, in addition to the initial LIGO and Virgo detectors, also includes the GEO600 [119] and TAMA300 [161] detectors, and several prototypes around the world. The impact of these projects onto the field is two-fold. First of all, the direct detection of GWs will be a milestone in science opening a new window to our universe, and marking the beginning of a new era in observational astronomy. Second, several groups around the world have already started to adapt the technology to novel interferometer concepts [60, 155], with potential applications not only in GW science, but also geophysics. The basic measurement scheme is always the same: the relative displacement of test masses is monitored by using ultra-stable lasers. Progress in this field is strongly dependent on how well the motion of the test masses can be shielded from the environment. Test masses are placed in vacuum and are either freely falling (e.g., atom clouds [137]), or suspended and seismically isolated (e.g., high-quality glass or crystal mirrors as used in all of the detectors listed above). The best seismic isolations realized so far are effective above a few Hz, which limits the frequency range of detectable gravity fluctuations. Nonetheless, low-frequency concepts are continuously improving, and it is conceivable that future detectors will be sufficiently sensitive to detect GWs well below a Hz [88].
Terrestrial gravity perturbations were identified as a potential noise source already in the first concept laid out for a laser-interferometric GW detector [171]. Today, this form of noise is known as “terrestrial gravitational noise”, “Newtonian noise”, or “gravity-gradient noise”. It has never been observed in GW detectors, but it is predicted to limit the sensitivity of the advanced GW detectors at low frequencies. The most important source of gravity noise comes from fluctuating seismic fields [151]. Gravity perturbations from atmospheric disturbances such as pressure and temperature fluctuations can become significant at lower frequencies [51]. Anthropogenic sources of gravity perturbations are easier to avoid, but could also be relevant at lower frequencies [163]. Today, we only have one example of a direct observation of gravity fluctuations, i.e., from pressure fluctuations of the atmosphere in high-precision gravimeters [128]. Therefore, almost our entire understanding of gravity fluctuations is based on models. Nonetheless, potential sensitivity limits of future large-scale GW detectors need to be identified and characterized well in advance, and so there is a need to continuously improve our understanding of terrestrial gravity noise. Based on our current understanding, the preferred option is to construct future GW detectors underground to avoid the most dominant Newtonian-noise contributions. This choice was made for the next-generation Japanese GW detector KAGRA, which is currently being constructed underground at the Kamioka site [17], and also as part of a design study for the Einstein Telescope in Europe [140, 139]. While the benefit from underground construction with respect to gravity noise is expected to be substantial in GW detectors sensitive above a few Hz [27], it can be argued that it is less effective at lower frequencies [88].
Alternative mitigation strategies includes coherent noise cancellation [42]. The idea is to monitor the sources of gravity perturbations using auxiliary sensors such as microphones and seismometers, and to use their data to generate a coherent prediction of gravity noise. This technique is successfully applied in gravimeters to reduce the foreground of atmospheric gravity noise using collocated pressure sensors [128]. It is also noteworthy that the models of the atmospheric gravity noise are consistent with observations. This should give us some confidence at least that coherent Newtonian-noise cancellation can also be achieved in GW detectors. It is evident though that a model-based prediction of the performance of coherent noise cancellation schemes is prone to systematic errors as long as the properties of the sources are not fully understood. Ongoing experiments at the Sanford Underground Research Facility with the goal to characterize seismic fields in three dimensions are expected to deliver first data from an underground seismometer array in 2015 (see [89] for results from an initial stage of the experiment). While most people would argue that constructing GW detectors underground is always advantageous, it is still necessary to estimate how much is gained and whether the science case strongly profits from it. This is a complicated problem that needs to be answered as part of a site selection process.
More recently, high-precision gravity strainmeters have been considered as monitors of geophysical signals [83]. Analytical models have been calculated, which allow us to predict gravity transients from seismic sources such as earthquakes. It was suggested to implement gravity strainmeters in existing earthquake-early warning systems to increase warning times. It is also conceivable that an alternative method to estimate source parameters using gravity signals will improve our understanding of seismic sources. Potential applications must still be investigated in greater detail, but the study already demonstrates that the idea to use GW technology to realize new geophysical sensors seems feasible. As explained in [49], gravitational forces start to dominate the dynamics of seismic phenomena below about 1 mHz (which coincides approximately with a similar transition in atmospheric dynamics where gravity waves start to dominate over other forms of oscillations [164]). Seismic isolation would be ineffective below 1 mHz since the gravitational acceleration of a test mass produced by seismic displacement becomes comparable to the seismic acceleration itself. Therefore, we claim that 10 mHz is about the lowest frequency at which ground-based gravity strainmeters will ever be able to detect GWs, and consequently, modelling terrestrial gravity perturbations in these detectors can focus on frequencies above 10 mHz.
This article is divided into six main sections. Section 2 serves as an introduction to gravity measurements focussing on the response mechanisms and basic properties of gravity sensors. Section 3 describes models of gravity perturbations from ambient seismic fields. The results can be used to estimate noise spectra at the surface and underground. A subsection is devoted to the problem of noise estimation in low-frequency GW detectors, which differs from high-frequency estimates mostly in that gravity perturbations are strongly correlated between different test masses. In the low-frequency regime, the gravity noise is best described as gravity-gradient noise. Section 4 is devoted to time domain models of transient gravity perturbations from seismic point sources. The formalism is applied to point forces and shear dislocations. The latter allows us to estimate gravity perturbations from earthquakes. Atmospheric models of gravity perturbations are presented in Section 5. This includes gravity perturbations from atmospheric temperature fields, infrasound fields, shock waves, and acoustic noise from turbulence. The solution for shock waves is calculated in time domain using the methods of Section 4. A theoretical framework to calculate gravity perturbations from objects is given in Section 6. Since many different types of objects can be potential sources of gravity perturbations, the discussion focusses on the development of a general method instead of summarizing all of the calculations that have been done in the past. Finally, Section 7 discusses possible passive and active noise mitigation strategies. Due to the complexity of the problem, most of the section is devoted to active noise cancellation providing the required analysis tools and showing limitations of this technique. Site selection is the main topic under passive mitigation, and is discussed in the context of reducing environmental noise and criteria relevant to active noise cancellation. Each of these sections ends with a summary and a discussion of open problems. While this article is meant to be a review of the current state of the field, it also presents new analyses especially with respect to the impact of seismic scattering on gravity perturbations (Sections 3.3.2 and 3.3.3), active gravity noise cancellation (Section 7.1.3), and timedomain models of gravity perturbations from atmospheric and seismic point sources (Sections 4.1, 4.5, and 5.3).
Even though evident to experts, it is worth emphasizing that all calculations carried out in this article have a common starting point, namely Newton’s universal law of gravitation. It states that the attractive gravitational force equation M1 between two point masses m1, m2 is given by
equation M21
where G = 6.672 × 10−11 N m2/kg2 is the gravitational constant. Eq. (1) gives rise to many complex phenomena on Earth such as inner-core oscillations [156], atmospheric gravity waves [157], ocean waves [94, 177], and co-seismic gravity changes [122]. Due to its importance, we will honor the eponym by referring to gravity noise as Newtonian noise in the following. It is thereby clarified that the gravity noise models considered in this article are non-relativistic, and propagation effects of gravity changes are neglected. While there could be interesting scenarios where this approximation is not fully justified (e.g., whenever a gravity perturbation can be sensed by several sensors and differences in arrival times can be resolved), it certainly holds in any of the problems discussed in this article. We now invite the reader to enjoy the rest of the article, and hope that it proves to be useful.
Go to:
Gravity Measurements
In this section, we describe the relevant mechanisms by which a gravity sensor can couple to gravity perturbations, and give an overview of the most widely used measurement schemes: the (relative) gravimeter [53, 181], the gravity gradiometer [125], and the gravity strainmeter. The last category includes the large-scale GW detectors Virgo [6], LIGO [91], GEO600 [119], KAGRA [17], and a new generation of torsion-bar antennas currently under development [13]. Also atom interferometers can potentially be used as gravity strainmeters in the future [62]. Strictly speaking, none of the sensors only responds to a single field quantity (such as changes in gravity acceleration or gravity strain), but there is always a dominant response mechanism in each case, which justifies to give the sensor a specific name. A clear distinction between gravity gradiometers and gravity strainmeters has never been made to our knowledge. Therefore the sections on these two measurement principles will introduce a definition, and it is by no means the only possible one. Later on in this article, we almost exclusively discuss gravity models relevant to gravity strainmeters since the focus lies on gravity fluctuations above 10 mHz. Today, the sensitivity near 10 mHz of gravimeters towards gravity fluctuations is still competitive to or exceeds the sensitivity of gravity strainmeters, but this is likely going to change in the future so that we can expect strainmeters to become the technology of choice for gravity observations above 10 mHz [88]. The following sections provide further details on this statement. Space-borne gravity experiments such as GRACE [167] will not be included in this overview. The measurement principle of GRACE is similar to that of gravity strainmeters, but only very slow changes of Earth gravity field can be observed, and for this reason it is beyond the scope of this article.
The different response mechanisms to terrestrial gravity perturbations are summarized in Section 2.1. While we will identify the tidal forces acting on the test masses as dominant coupling mechanism, other couplings may well be relevant depending on the experiment. The Shapiro time delay will be discussed as the only relativistic effect. Higher-order relativistic effects are neglected. All other coupling mechanisms can be calculated using Newtonian theory including tidal forces, coupling in static non-uniform gravity fields, and coupling through ground displacement induced by gravity fluctuations. In Sections 2.2 to 2.4, the different measurement schemes are explained including a brief summary of the sensitivity limitations (choosing one of a few possible experimental realizations in each case). As mentioned before, we will mostly develop gravity models relevant to gravity strainmeters in the remainder of the article. Therefore, the detailed discussion of alternative gravimetry concepts mostly serves to highlight important differences between these concepts, and to develop a deeper understanding of the instruments and their role in gravity measurements.
Gravity response mechanisms
Gravity acceleration and tidal forces We will start with the simplest mechanism of all, the acceleration of a test mass in the gravity field. Instruments that measure the acceleration are called gravimeters. A test mass inside a gravimeter can be freely falling such as atom clouds [181] or, as suggested as possible future development, even macroscopic objects [72]. Typically though, test masses are supported mechanically or magnetically constraining motion in some of its degrees of freedom. A test mass suspended from strings responds to changes in the horizontal gravity acceleration. A test mass attached at the end of a cantilever with horizontal equilibrium position responds to changes in vertical gravity acceleration. The support fulfills two purposes. First, it counteracts the static gravitational force in a way that the test mass can respond to changes in the gravity field along a chosen degree of freedom. Second, it isolates the test mass from vibrations. Response to signals and isolation performance depend on frequency. If the support is modelled as a linear, harmonic oscillator, then the test mass response to gravity changes extends over all frequencies, but the response is strongly suppressed below the oscillators resonance frequency. The response function between the gravity perturbation δg(ω) and induced test mass acceleration δa(ω) assumes the form
equation M32
where we have introduced a viscous damping parameter γ, and ω0 is the resonance frequency. Well below resonance, the response is proportional to ω2, while it is constant well above resonance. Above resonance, the supported test mass responds like a freely falling mass, at least with respect to “soft” directions of the support. The test-mass response to vibrations δα(ω) of the support is given by
equation M43
This applies for example to horizontal vibrations of the suspension points of strings that hold a test mass, or to vertical vibrations of the clamps of a horizontal cantilever with attached test mass. Well above resonance, vibrations are suppressed by ω−2, while no vibration isolation is provided below resonance. The situation is somewhat more complicated in realistic models of the support especially due to internal modes of the mechanical system (see for example [76]), or due to coupling of degrees of freedom [121]. Large mechanical support structures can feature internal resonances at relatively low frequencies, which can interfere to some extent with the desired performance of the mechanical support [173]. While Eqs. (2) and (3) summarize the properties of isolation and response relevant for this paper, details of the readout method can fundamentally impact an instrument’s response to gravity fluctuations and its susceptibility to seismic noise, as explained in Sections 2.2 to 2.4.
Next, we discuss the response to tidal forces. In Newtonian theory, tidal forces cause a relative acceleration δg12(ω) between two freely falling test masses according to
equation M54
where equation M6 is the Fourier amplitude of the gravity potential. The last equation holds if the distance r12 between the test masses is sufficiently small, which also depends on the frequency. The term equation M7 is called gravity-gradient tensor. In Newtonian approximation, the second time integral of this tensor corresponds to gravity strain equation M8, which is discussed in more detail in Section 2.4. Its trace needs to vanish in empty space since the gravity potential fulfills the Poisson equation. Tidal forces produce the dominant signals in gravity gradiometers and gravity strainmeters, which measure the differential acceleration or associated relative displacement between two test masses (see Sections 2.3 and 2.4). If the test masses used for a tidal measurement are supported, then typically the supports are designed to be as similar as possible, so that the response in Eq. (2) holds for both test masses approximately with the same parameter values for the resonance frequencies (and to a lesser extent also for the damping). For the purpose of response calibration, it is less important to know the parameter values exactly if the signal is meant to be observed well above the resonance frequency where the response is approximately equal to 1 independent of the resonance frequency and damping (here, “well above” resonance also depends on the damping parameter, and in realistic models, the signal frequency also needs to be “well below” internal resonances of the mechanical support).
Shapiro time delay Another possible gravity response is through the Shapiro time delay [19]. This effect is not universally present in all gravity sensors, and depends on the readout mechanism. Today, the best sensitivities are achieved by reflecting laser beams from test masses in interferometric configurations. If the test mass is displaced by gravity fluctuations, then it imprints a phase shift onto the reflected laser, which can be observed in laser interferometers, or using phasemeters. We will give further details on this in Section 2.4. In Newtonian gravity, the acceleration of test masses is the only predicted response to gravity fluctuations. However, from general relativity we know that gravity also affects the propagation of light. The leading-order term is the Shapiro time delay, which produces a phase shift of the laser beam with respect to a laser propagating in flat space. It can be calculated from the weak-field spacetime metric (see chapter 18 in [124]):
equation M95
Here, c is the speed of light, ds is the so-called line element of a path in spacetime, and equation M10. Additionally, for this metric to hold, motion of particles in the source of the gravity potential responsible for changes of the gravity potential need to be much slower than the speed of light, and also stresses inside the source must be much smaller than its mass energy density. All conditions are fulfilled in the case of Earth gravity field. Light follows null geodesics with ds2 = 0. For the spacetime metric in Eq. (5), we can immediately write
equation M116
As we will find out, this equation can directly be used to calculate the time delay as an integral along a straight line in terms of the coordinates equation M12, but this is not immediately clear since light bends in a gravity field. So one may wonder if integration along the proper light path instead of a straight line yields additional significant corrections. The so-called geodesic equation must be used to calculate the path. It is a set of four differential equations, one for each coordinate t, equation M13 in terms of a parameter λ. The weak-field geodesic equation is obtained from the metric in Eq. (5):
equation M147
where we have made use of Eq. (6) and the slow-motion condition equation M15. The coordinates equation M16 are to be understood as functions of λ. Since the deviation of a straight path is due to a weak gravity potential, we can solve these equations by perturbation theory introducing expansions equation M17 and t = t(0) +t(1) + …. The superscript indicates the order in ψ/c2. The unperturbed path has the simple parametrization
equation M188
We have chosen integration constants such that unperturbed time t(0) and parameter λ can be used interchangeably (apart from a shift by t0). Inserting these expressions into the right-hand side of Eq. (7), we obtain
equation M199
As we can see, up to linear order in equation M20, the deviation equation M21 is in orthogonal direction to the unperturbed path equation M22, which means that the deviation can be neglected in the calculation of the time delay. After some transformations, it is possible to derive Eq. (6) from Eq. (9), and this time we find explicitly that the right-hand-side of the equation only depends on the unperturbed coordinates1. In other words, we can integrate the time delay along a straight line as defined in Eq. (8), and so the total phase integrated over a travel distance L is given by
equation M2310
In static gravity fields, the phase shift doubles if the light is sent back since not only the direction of integration changes, but also the sign of the expression substituted for dt/dλ.
Gravity induced ground motion As we will learn in Section 3, seismic fields produce gravity perturbations either through density fluctuations of the ground, or by displacing interfaces between two materials of different density. It is also well-known in seismology that seismic fields can be affected significantly by self-gravity. Self-gravity means that the gravity perturbation produced by a seismic field acts back on the seismic field. The effect is most significant at low frequency where gravity induced acceleration competes against acceleration from elastic forces. In seismology, low-frequency seismic fields are best described in terms of Earth’s normal modes [55]. Normal modes exist as toroidal modes and spheroidal modes. Spheroidal modes are influenced by self-gravity, toroidal modes are not. For example, predictions of frequencies and shapes of spheroidal modes based on Earth models such as PREM (Preliminary Reference Earth Model) [68] are inaccurate if self-gravity effects are excluded. What this practically means is that in addition to displacement amplitudes, gravity becomes a dynamical variable in the elastodynamic equations that determine the normal-mode properties. Therefore, seismic displacement and gravity perturbation cannot be separated in normal-mode formalism (although self-gravity can be neglected in calculations of spheroidal modes at sufficiently high frequency).
In certain situations, it is necessary or at least more intuitive to separate gravity from seismic fields. An exotic example is Earth’s response to GWs [67, 49, 47, 30, 48]. Another example is the seismic response to gravity perturbations produced by strong seismic events at large distance to the source as described in Section 4. It is more challenging to analyze this scenario using normal-mode formalism. The sum over all normal modes excited by the seismic event (each of which describing a global displacement field) must lead to destructive interference of seismic displacement at large distances (where seismic waves have not yet arrived), but not of the gravity amplitudes since gravity is immediately perturbed everywhere. It can be easier to first calculate the gravity perturbation from the seismic perturbation, and then to calculate the response of the seismic field to the gravity perturbation at larger distance. This method will be adopted in this section. Gravity fields will be represented as arbitrary force or tidal fields (detailed models are presented in later sections), and we simply calculate the response of the seismic field. Normal-mode formalism can be avoided only at sufficiently high frequencies where the curvature of Earth does not significantly influence the response (i.e., well above 10 mHz). In this section, we will model the ground as homogeneous half space, but also more complex geologies can in principle be assumed.
Gravity can be introduced in two ways into the elastodynamic equations, as a conservative force −∇ψ [146, 169], or as tidal strain The latter method was described first by Dyson to calculate Earth’s response to GWs [67]. The approach also works for Newtonian gravity, with the difference that the tidal field produced by a GW is necessarily a quadrupole field with only two degrees of freedom (polarizations), while tidal fields produced by terrestrial sources are less constrained. Certainly, GWs can only be fully described in the framework of general relativity, which means that their representation as a Newtonian tidal field cannot be used to explain all possible observations [124]. Nonetheless, important here is that Dyson’s method can be extended to Newtonian tidal fields. Without gravity, the elastodynamic equations for small seismic displacement can be written as
equation M2411
where equation M25 is the seismic displacement field, and equation M26 is the stress tensor [9]. In the absence of other forces, the stress is determined by the seismic field. In the case of a homogeneous and isotropic medium, the stress tensor for small seismic displacement can be written as
equation M2712
The quantity equation M28 is known as seismic strain tensor, and λ, μ are the Lamé constants (see Section 3.1). Its trace is equal to the divergence of the displacement field. Dyson introduced the tidal field from first principles using Lagrangian mechanics, but we can follow a simpler approach. Eq. (12) means that a stress field builds up in response to a seismic strain field, and the divergence of the stress field acts as a force producing seismic displacement. The same happens in response to a tidal field, which we represent as gravity strain equation M29. A strain field changes the distance between two freely falling test masses separated by equation M30 by equation M312. For sufficiently small distances L, the strain field can be substituted by the second time integral of the gravity-gradient tensor equation M32. If the masses are not freely falling, then the strain field acts as an additional force. The corresponding contribution to the material’s stress tensor can be written
equation M3313
Since we assume that the gravity field is produced by a distant source, the local contribution to gravity perturbations is neglected, which means that the gravity potential obeys the Laplace equation, equation M34. Calculating the divergence of the stress tensor according to Eq. (11), we find that the gravity term vanishes! This means that a homogeneous and isotropic medium does not respond to gravity strain fields. However, we have to be more careful here. Our goal is to calculate the response of a half-space to gravity strain. Even if the half-space is homogeneous, the Lamé constants change discontinuously across the surface. Hence, at the surface, the divergence of the stress tensor reads
equation M3514
In other words, tidal fields produce a force onto an elastic medium via gradients in the shear modulus (second Lamé constant). The gradient of the shear modulus can be written in terms of a Dirac delta function, equation M36, for a flat surface at z = 0 with unit normal vector equation M37. The response to gravity strain fields is obtained applying the boundary condition of vanishing surface traction, equation M38:
equation M3915
Once the seismic strain field is calculated, it can be used to obtain the seismic stress, which determines the displacement field equation M40 according to Eq. (11). In this way, one can for example calculate that a seismometer or gravimeter can observe GWs by monitoring surface displacement as was first calculated by Dyson [67].
Coupling in non-uniform, static gravity fields If the gravity field is static, but non-uniform, then displacement equation M41 of the test mass in this field due to a non-gravitational fluctuating force is associated with a changing gravity acceleration according to
equation M4216
We introduce a characteristic length λ, over which gravity acceleration varies significantly. Hence, we can rewrite the last equation in terms of the associated test-mass displacement ζ
equation M4317
where we have neglected directional dependence and numerical factors. The acceleration change from motion in static, inhomogeneous fields is generally more significant at low frequencies. Let us consider the specific case of a suspended test mass. It responds to fluctuations in horizontal gravity acceleration. The test mass follows the motion of the suspension point in vertical direction (i.e., no seismic isolation), while seismic noise in horizontal direction is suppressed according to Eq. (3). Accordingly, it is possible that the unsuppressed vertical (z-axis) seismic noise ξz(t) coupling into the horizontal (x-axis) motion of the test mass through the term ∂xgz = ∂zgx dominates over the gravity response term in Eq. (2). Due to additional coupling mechanisms between vertical and horizontal motion in real seismic-isolation systems, test masses especially in GW detectors are also isolated in vertical direction, but without achieving the same noise suppression as in horizontal direction. For example, the requirements on vertical test-mass displacement for Advanced LIGO are a factor 1000 less stringent than on the horizontal displacement [22]. Requirements can be set on the vertical isolation by estimating the coupling of vertical motion into horizontal motion, which needs to take the gravity-gradient coupling of Eq. (16) into account. Although, because of the frequency dependence, gravity-gradient effects are more significant in low-frequency detectors, such as the space-borne GW detector LISA [154].
Next, we calculate an estimate of gravity gradients in the vicinity of test masses in large-scale GW detectors, and see if the gravity-gradient coupling matters compared to mechanical vertical-to-horizontal coupling.
One contribution to gravity gradients will come from the vacuum chamber surrounding the test mass. We approximate the shape of the chamber as a hollow cylinder with open ends (open ends just to simplify the calculation). In our calculation, the test mass can be offset from the cylinder axis and be located at any distance to the cylinder ends (we refer to this coordinate as height). The gravity field can be expressed in terms of elliptic integrals, but the explicit solution is not of concern here. Instead, let us take a look at the results in Figure Figure1.1. Gravity gradients ∂zgx vanish if the test mass is located on the symmetry axis or at height L/2. There are also two additional ∂zgx = 0 contour lines starting at the symmetry axis at heights ∼ 0.24 and ∼0.76. Let us assume that the test mass is at height 0.3L, a distance 0.05L from the cylinder axis, the total mass of the cylinder is M = 5000 kg, and the cylinder height is L = 4 m. In this case, the gravity-gradient induced vertical-to-horizontal coupling factor at 20 Hz is
equation M4418
This means that gravity-gradient induced coupling is extremely weak, and lies well below estimates of mechanical coupling (of order 0.001 in Advanced LIGO3). Even though the vacuum chamber was modelled with a very simple shape, and additional asymmetries in the mass distribution around the test mass may increase gravity gradients, it still seems very unlikely that the coupling would be significant. As mentioned before, one certainly needs to pay more attention when calculating the coupling at lower frequencies. The best procedure is of course to have a 3D model of the near test-mass infrastructure available and to use it for a precise calculation of the gravity-gradient field.
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Figure 1
Gravity gradients inside hollow cylinder. The total height of the cylinder is L, and M is its total mass. The radius of the cylinder is 0.3L. The axes correspond to the distance of the test mass from the symmetry axis of the cylinder, and its height above one of the cylinders ends. The plot on the right is simply a zoom of the left plot into the intermediate heights.
Gravimeters
Gravimeters are instruments that measure the displacement of a test mass with respect to a non-inertial reference rigidly connected to the ground. The test mass is typically supported mechanically or magnetically (atom-interferometric gravimeters are an exception), which means that the test-mass response to gravity is altered with respect to a freely falling test mass. We will use Eq. (2) as a simplified response model. There are various possibilities to measure the displacement of a test mass. The most widespread displacement sensors are based on capacitive readout, as for example used in superconducting gravimeters (see Figure Figure22 and [96]). Sensitive displacement measurements are in principle also possible with optical readout systems; a method that is (necessarily) implemented in atom-interferometric gravimeters [137], and prototype seismometers [34] (we will explain the distinction between seismometers and gravimeters below). As will become clear in Section 2.4, optical readout is better suited for displacement measurements over long baselines, as required for the most sensitive gravity strain measurements, while the capacitive readout should be designed with the smallest possible distance between the test mass and the non-inertial reference [104].
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Figure 2
Sketch of a levitated sphere serving as test mass in a superconducting gravimeter. Dashed lines indicate magnetic field lines. Coils are used for levitation and precise positioning of the sphere. Image reproduced with permission from [96]; copyright by Elsevier.
Let us take a closer look at the basic measurement scheme of a superconducting gravimeter shown in Figure Figure2.2. The central part is formed by a spherical superconducting shell that is levitated by superconducting coils. Superconductivity provides stability of the measurement, and also avoids some forms of noise (see [96] for details). In this gravimeter design, the lower coil is responsible mostly to balance the mean gravitational force acting on the sphere, while the upper coil modifies the magnetic gradient such that a certain “spring constant” of the magnetic levitation is realized. In other words, the current in the upper coil determines the resonance frequency in Eq. (2).
Capacitor plates are distributed around the sphere. Whenever a force acts on the sphere, the small signal produced in the capacitive readout is used to immediately cancel this force by a feedback coil. In this way, the sphere is kept at a constant location with respect to the external frame. This illustrates a common concept in all gravimeters. The displacement sensors can only respond to relative displacement between a test mass and a surrounding structure. If small gravity fluctuations are to be measured, then it is not sufficient to realize low-noise readout systems, but also vibrations of the surrounding structure forming the reference frame must be as small as possible. In general, as we will further explore in the coming sections, gravity fluctuations are increasingly dominant with decreasing frequency. At about 1 mHz, gravity acceleration associated with fluctuating seismic fields become comparable to seismic acceleration, and also atmospheric gravity noise starts to be significant [53]. At higher frequencies, seismic acceleration is much stronger than typical gravity fluctuations, which means that the gravimeter effectively operates as a seismometer. In summary, at sufficiently low frequencies, the gravimeter senses gravity accelerations of the test mass with respect to a relatively quiet reference, while at higher frequencies, the gravimeter senses seismic accelerations of the reference with respect to a test mass subject to relatively small gravity fluctuations. In superconducting gravimeters, the third important contribution to the response is caused by vertical motion ξ(t) of a levitated sphere against a static gravity gradient (see Section 2.1.4). As explained above, feedback control suppresses relative motion between sphere and gravimeter frame, which causes the sphere to move as if attached to the frame or ground. In the presence of a static gravity gradient ∂zgz, the motion of the sphere against this gradient leads to a change in gravity, which alters the feedback force (and therefore the recorded signal). The full contribution from gravitational, δa(t), and seismic, equation M45, accelerations can therefore be written
equation M4619
It is easy to verify, using Eqs. (2) and (3), that the relative amplitude of gravity and seismic fluctuations from the first two terms is independent of the test-mass support. Therefore, vertical seismic displacement of the reference frame must be considered fundamental noise of gravimeters and can only be avoided by choosing a quiet measurement site. Obviously, Eq. (19) is based on a simplified support model. One of the important design goals of the mechanical support is to minimize additional noise due to non-linearities and cross-coupling. As is explained further in Section 2.3, it is also not possible to suppress seismic noise in gravimeters by subtracting the disturbance using data from a collocated seismometer. Doing so inevitably turns the gravimeter into a gravity gradiometer.
Gravimeters target signals that typically lie well below 1 mHz. Mechanical or magnetic supports of test masses have resonance frequencies at best slightly below 10 mHz along horizontal directions, and typically above 0.1 Hz in the vertical direction [23, 174]4. Well below resonance frequency, the response function can be approximated as equation M47. At first, it may look as if the gravimeter should not be sensitive to very low-frequency fluctuations since the response becomes very weak. However, the strength of gravity fluctuations also strongly increases with decreasing frequency, which compensates the small response. It is clear though that if the resonance frequency was sufficiently high, then the response would become so weak that the gravity signal would not stand out above other instrumental noise anymore. The test-mass support would be too stiff. The sensitivity of the gravimeter depends on the resonance frequency of the support and the intrinsic instrumental noise. With respect to seismic noise, the stiffness of the support has no influence as explained before (the test mass can also fall freely as in atom interferometers).
For superconducting gravimeters of the Global Geodynamics Project (GGP) [52], the median spectra are shown in Figure Figure3.3. Between 0.1 mHz and 1 mHz, atmospheric gravity perturbations typically dominate, while instrumental noise is the largest contribution between 1 mHz and 5 mHz [96]. The smallest signal amplitudes that have been measured by integrating long-duration signals is about 10−12 m/s2. A detailed study of noise in superconducting gravimeters over a larger frequency range can be found in [145]. Note that in some cases, it is not fit to categorize seismic and gravity fluctuations as noise and signal. For example, Earth’s spherical normal modes coherently excite seismic and gravity fluctuations, and the individual contributions in Eq. (19) have to be understood only to accurately translate data into normal-mode amplitudes [55].
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Figure 3
Median spectra of superconducting gravimeters of the GGP. Image reproduced with permission from [48]; copyright by APS.
Gravity gradiometers
It is not the purpose of this section to give a complete overview of the different gradiometer designs. Gradiometers find many practical applications, for example in navigation and resource exploration, often with the goal to measure static or slowly changing gravity gradients, which do not concern us here. For example, we will not discuss rotating gradiometers, and instead focus on gradiometers consisting of stationary test masses. While the former are ideally suited to measure static or slowly changing gravity gradients with high precision especially under noisy conditions, the latter design has advantages when measuring weak tidal fluctuations. In the following, we only refer to the stationary design. A gravity gradiometer measures the relative acceleration between two test masses each responding to fluctuations of the gravity field [102, 125]. The test masses have to be located close to each other so that the approximation in Eq. (4) holds. The proximity of the test masses is used here as the defining property of gradiometers. They are therefore a special type of gravity strainmeter (see Section 2.4), which denotes any type of instrument that measures relative gravitational acceleration (including the even more general concept of measuring space-time strain).
Gravity gradiometers can be realized in two versions. First, one can read out the position of two test masses with respect to the same rigid, non-inertial reference. The two channels, each of which can be considered a gravimeter, are subsequently subtracted. This scheme is for example realized in dual-sphere designs of superconducting gravity gradiometers [90] or in atom-interferometric gravity gradiometers [159].
It is schematically shown in Figure Figure4.4. Let us first consider the dual-sphere design of a superconducting gradiometer. If the reference is perfectly stiff, and if we assume as before that there are no cross-couplings between degrees of freedom and the response is linear, then the subtraction of the two gravity channels cancels all of the seismic noise, leaving only the instrumental noise and the differential gravity signal given by the second line of Eq. (4). Even in real setups, the reduction of seismic noise can be many orders of magnitude since the two spheres are close to each other, and the two readouts pick up (almost) the same seismic noise [125]. This does not mean though that gradiometers are necessarily more sensitive instruments to monitor gravity fields. A large part of the gravity signal (the common-mode part) is subtracted together with the seismic noise, and the challenge is now passed from finding a seismically quiet site to developing an instrument with lowest possible intrinsic noise.
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Figure 4
Basic scheme of a gravity gradiometer for measurements along the vertical direction. Two test masses are supported by horizontal cantilevers (superconducting magnets, …). Acceleration of both test masses is measured against the same non-inertial reference frame, which is connected to the ground. Each measurement constitutes one gravimeter. Subtraction of the two channels yields a gravity gradiometer.
The atom-interferometric gradiometer differs in some important details from the superconducting gradiometer. The test masses are realized by ultracold atom clouds, which are (nearly) freely falling provided that magnetic shielding of the atoms is sufficient, and interaction between atoms can be neglected. Interactions of a pair of atom clouds with a laser beam constitute the basic gravity gradiometer scheme. Even though the test masses are freely falling, the readout is not generally immune to seismic noise [80, 18]. The laser beam interacting with the atom clouds originates from a source subject to seismic disturbances, and interacts with optics that require seismic isolation. Schemes have been proposed that could lead to a large reduction of seismic noise [178, 77], but their effectiveness has not been tested in experiments yet. Since the differential position (or tidal) measurement is performed using a laser beam, the natural application of atom-interferometer technology is as gravity strainmeter (as explained before, laser beams are favorable for differential position measurements over long baselines). Nonetheless, the technology is currently insufficiently developed to realize large-baseline experiments, and we can therefore focus on its application in gradiometry. Let us take a closer look at the response of atom-interferometric gradiometers to seismic noise. In atom-interferometric detectors (excluding the new schemes proposed in [178, 77]), one can show that seismic acceleration δα(ω) of the optics or laser source limits the sensitivity of a tidal measurement according to
equation M4820
where L is the separation of the two atom clouds, and is the speed of light. It should be emphasized that the seismic noise remains, even if all optics and the laser source are all linked to the same infinitely stiff frame. In addition to this noise term, other coupling mechanisms may play a role, which can however be suppressed by engineering efforts. The noise-reduction factor ωL/c needs to be compared with the common-mode suppression of seismic noise in superconducting gravity gradiometers, which depends on the stiffness of the instrument frame, and on contamination from cross coupling of degrees-of-freedom. While the seismic noise in Eq. (20) is a fundamental noise contribution in (conventional) atom-interferometric gradiometers, the noise suppression in superconducting gradiometers depends more strongly on the engineering effort (at least, we venture to claim that common-mode suppression achieved in current instrument designs is well below what is fundamentally possible).
To conclude this section, we discuss in more detail the connection between gravity gradiometers and seismically (actively or passively) isolated gravimeters. As we have explained in Section 2.2, the sensitivity limitation of gravimeters by seismic noise is independent of the mechanical support of the test mass (assuming an ideal, linear support). The main purpose of the mechanical support is to maximize the response of the test mass to gravity fluctuations, and thereby increase the signal with respect to instrumental noise other than seismic noise. Here we will explain that even a seismic isolation of the gravimeter cannot overcome this noise limitation, at least not without fundamentally changing its response to gravity fluctuations. Let us first consider the case of a passively seismically isolated gravimeter. For example, we can imagine that the gravimeter is suspended from the tip of a strong horizontal cantilever. The system can be modelled as two oscillators in a chain, with a light test mass m supported by a heavy mass M representing the gravimeter (reference) frame, which is itself supported from a point rigidly connected to Earth. The two supports are modelled as harmonic oscillators. As before, we neglect cross coupling between degrees of freedom. Linearizing the response of the gravimeter frame and test mass for small accelerations, and further neglecting terms proportional to m/M, one finds the gravimeter response to gravity fluctuations:
equation M4921
Here, ω1, γ1 are the resonance frequency and damping of the gravimeter support, while ω2, γ2 are the resonance frequency and damping of the test-mass support. The response and isolation functions R(·), S(·) are defined in Eqs. (2) and (3). Remember that Eq. (21) is obtained as a differential measurement of test-mass acceleration versus acceleration of the reference frame. Therefore, δg1(ω) denotes the gravity fluctuation at the center-of-mass of the gravimeter frame, and δg2(ω) at the test mass. An infinitely stiff gravimeter suspension, ω1 → ∞, yields R(ω; ω1, γ1) = 0, and the response turns into the form of the non-isolated gravimeter. The seismic isolation is determined by
equation M5022
We can summarize the last two equations as follows. At frequencies well above ω1, the seismically isolated gravimeter responds like a gravity gradiometer, and seismic noise is strongly suppressed. The deviation from the pure gradiometer response ∼ δg2(ω) − δg1(ω) is determined by the same function S(ω; ω1, γ1) that describes the seismic isolation. In other words, if the gravity gradient was negligible, then we ended up with the conventional gravimeter response, with signals suppressed by the seismic isolation function. Well below ω1, the seismically isolated gravimeter responds like a conventional gravimeter without seismic-noise reduction. If the centers of the masses m (test mass) and M (reference frame) coincide, and therefore δg1(ω) = δg2(ω), then the response is again like a conventional gravimeter, but this time suppressed by the isolation function S(ω; ω1, γ1).
Let us compare the passively isolated gravimeter with an actively isolated gravimeter. In active isolation, the idea is to place the gravimeter on a stiff platform whose orientation can be controlled by actuators. Without actuation, the platform simply follows local surface motion. There are two ways to realize an active isolation. One way is to place a seismometer next to the platform onto the ground, and use its data to subtract ground motion from the platform. The actuators cancel the seismic forces. This scheme is called feed-forward noise cancellation. Feed-forward cancellation of gravity noise is discussed at length in Section 7.1, which provides details on its implementation and limitations. The second possibility is to place the seismometer together with the gravimeter onto the platform, and to suppress seismic noise in a feedback configuration [4, 2]. In the following, we discuss the feed-forward technique as an example since it is easier to analyze (for example, feedback control can be unstable [4]). As before, we focus on gravity and seismic fluctuations. The seismometer’s intrinsic noise plays an important role in active isolation limiting its performance, but we are only interested in the modification of the gravimeter’s response. Since there is no fundamental difference in how a seismometer and a gravimeter respond to seismic and gravity fluctuations, we know from Section 2.2 that the seismometer output is proportional to δg1(ω) − δα(ω), i.e., using a single test mass for acceleration measurements, seismic and gravity perturbations contribute in the same way. A transfer function needs to be multiplied to the acceleration signals, which accounts for the mechanical support and possibly also electronic circuits involved in the seismometer readout. To cancel the seismic noise of the platform that carries the gravimeter, the effect of all transfer functions needs to be reversed by a matched feed-forward filter. The output of the filter is then equal to δg1(ω) − δα(ω) and is added to the motion of the platform using actuators cancelling the seismic noise and adding the seismometer’s gravity signal. In this case, the seismometer’s gravity signal takes the place of the seismic noise in Eq. (3). The complete gravity response of the actively isolated gravimeter then reads
equation M5123
The response is identical to a gravity gradiometer, where ω2, γ2 are the resonance frequency and damping of the gravimeter’s test-mass support. In reality, instrumental noise of the seismometer will limit the isolation performance and introduce additional noise into Eq. (23). Nonetheless, Eqs. (21) and (23) show that any form of seismic isolation turns a gravimeter into a gravity gradiometer at frequencies where seismic isolation is effective. For the passive seismic isolation, this means that the gravimeter responds like a gradiometer at frequencies well above the resonance frequency ω1 of the gravimeter support, while it behaves like a conventional gravimeter below ω1. From these results it is clear that the design of seismic isolations and the gravity response can in general not be treated independently. As we will see in Section 2.4 though, tidal measurements can profit strongly from seismic isolation especially when common-mode suppression of seismic noise like in gradiometers is insufficient or completely absent.
Gravity strainmeters
Gravity strain is an unusual concept in gravimetry that stems from our modern understanding of gravity in the framework of general relativity. From an observational point of view, it is not much different from elastic strain. Fluctuating gravity strain causes a change in distance between two freely falling test masses, while seismic or elastic strain causes a change in distance between two test masses bolted to an elastic medium. It should be emphasized though that we cannot always use this analogy to understand observations of gravity strain [106]. Fundamentally, gravity strain corresponds to a perturbation of the metric that determines the geometrical properties of spacetime [124]. We will briefly discuss GWs, before returning to a Newtonian description of gravity strain.
Gravitational waves are weak perturbations of spacetime propagating at the speed of light. Freely falling test masses change their distance in the field of a GW. When the length of the GW is much larger than the separation between the test masses, it is possible to interpret this change as if caused by a Newtonian force. We call this the long-wavelength regime. Since we are interested in the low-frequency response of gravity strainmeters throughout this article (i.e., frequencies well below 100 Hz), this condition is always fulfilled for Earth-bound experiments. The effect of a gravity-strain field equation M52 on a pair of test masses can then be represented as an equivalent Newtonian tidal field
equation M5324
Here, equation M54 is the relative acceleration between two freely falling test masses, L is the distance between them, and equation M55 is the unit vector pointing from one to the other test mass, and equation M56 its transpose. As can be seen, the gravity-strain field is represented by a 3 × 3 tensor. It contains the space-components of a 4-dimensional metric perturbation of spacetime, and determines all properties of GWs5. Note that the strain amplitude h in Eq. (24) needs to be multiplied by 2 to obtain the corresponding amplitude of the metric perturbation (e.g., the GW amplitude). Throughout this article, we define gravity strain as h = ΔL/L, while the effect of a GW with amplitude aGW on the separation of two test mass is determined by aGW = 2ΔL/L.
The strain field of a GW takes the form of a quadrupole oscillation with two possible polarizations commonly denoted × (cross)-polarization and +(plus)-polarization. The arrows in Figure Figure55 indicate the lines of the equivalent tidal field of Eq. (24).
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Figure 5
Polarizations of a gravitational wave.
Consequently, to (directly) observe GWs, one can follow two possible schemes: (1) the conventional method, which is a measurement of the relative displacement of suspended test masses typically carried out along two perpendicular baselines (arms); and (2) measurement of the relative rotation between two suspended bars. Figure Figure66 illustrates the two cases. In either case, the response of a gravity strainmeter is obtained by projecting the gravity strain tensor onto a combination of two unit vectors, equation M57 and equation M58, that characterize the orientation of the detector, such as the directions of two bars in a rotational gravity strain meter, or of two arms of a conventional gravity strain meter. This requires us to define two different gravity strain projections. The projection for the rotational strain measurement is given by
equation M5925
where the subscript × indicates that the detector responds to the ×-polarization assuming that the x, y-axes (see Figure Figure5)5) are oriented along two perpendicular bars. The vectors equation M60 and equation M61 are rotated counter-clockwise by 90° with respect to equation M62 and equation M63. In the case of perpendicular bars equation M64 and equation M65. The corresponding projection for the conventional gravity strain meter reads
equation M6626
The subscript + indicates that the detector responds to the +-polarization provided that the x, y-axes are oriented along two perpendicular baselines (arms) of the detector. The two schemes are shown in Figure Figure6.6. The most sensitive GW detectors are based on the conventional method, and distance between test masses is measured by means of laser interferometry. The LIGO and Virgo detectors have achieved strain sensitivities of better than 10−22 Hz−1/2 between about 50 Hz and 1000 Hz in past science runs and are currently being commissioned in their advanced configurations [91, 7]. The rotational scheme is realized in torsion-bar antennas, which are considered as possible technology for sub-Hz GW detection [155, 69]. However, with achieved strain sensitivity of about 10−8 Hz−1/2 near 0.1 Hz, the torsion-bar detectors are far from the sensitivity we expect to be necessary for GW detection [88].
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Figure 6
Sketches of the relative rotational and displacement measurement schemes.
Let us now return to the discussion of the previous sections on the role of seismic isolation and its impact on gravity response. Gravity strainmeters profit from seismic isolation more than gravimeters or gravity gradiometers. We have shown in Section 2.2 that seismically isolated gravimeters are effectively gravity gradiometers. So in this case, seismic isolation changes the response of the instrument in a fundamental way, and it does not make sense to talk of seismically isolated gravimeters. Seismic isolation could in principle be beneficial for gravity gradiometers (i.e., the acceleration of two test masses is measured with respect to a common rigid, seismically isolated reference frame), but the common-mode rejection of seismic noise (and gravity signals) due to the differential readout is typically so high that other instrumental noise becomes dominant. So it is possible that some gradiometers would profit from seismic isolation, but it is not generally true. Let us now consider the case of a gravity strainmeter. As explained in Section 2.3, we distinguish gradiometers and strainmeters by the distance of their test masses. For example, the distance of the LIGO or Virgo test masses is 4 km and 3 km respectively. Seismic noise and terrestrial gravity fluctuations are insignificantly correlated between the two test masses within the detectors’ most sensitive frequency band (above 10 Hz). Therefore, the approximation in Eq. (4) does not apply. Certainly, the distinction between gravity gradiometers and strainmeters remains somewhat arbitrary since at any frequency the approximation in Eq. (4) can hold for one type of gravity fluctuation, while it does not hold for another. Let us adopt a more practical definition at this point. Whenever the design of the instrument places the test masses as distant as possible from each other given current technology, then we call such an instrument strainmeter. In the following, we will discuss seismic isolation and gravity response for three strainmeter designs, the laser-interferometric, atom-interferometric, and superconducting strainmeters. It should be emphasized that the atom-interferometric and superconducting concepts are still in the beginning of their development and have not been realized yet with scientifically interesting sensitivities.
Laser-interferometric strainmeters The most sensitive gravity strainmeters, namely the large-scale GW detectors, use laser interferometry to read out the relative displacement between mirror pairs forming the test masses. Each test mass in these detectors is suspended from a seismically isolated platform, with the suspension itself providing additional seismic isolation. Section 2.1.1 introduced a simplified response and isolation model based on a harmonic oscillator characterized by a resonance frequency ω0 and viscous damping γ6. In a multi-stage isolation and suspension system as realized in GW detectors (see for example [37, 121]), coupling between multiple oscillators cannot be neglected, and is fundamental to the seismic isolation performance, but the basic features can still be explained with the simplified isolation and response model of Eqs. (2) and (3). The signal output of the interferometer is proportional to the relative displacement between test masses. Since seismic noise is approximately uncorrelated between two distant test masses, the differential measurement itself cannot reject seismic noise as in gravity gradiometers. Without seismic isolation, the dominant signal would be seismic strain, i.e., the distance change between test masses due to elastic deformation of the ground, with a value of about 10−15 Hz−1/2 at 50 Hz (assuming kilometer-scale arm lengths). At the same time, without seismically isolated test masses, the gravity signal can only come from the ground response to gravity fluctuations as described in Section 2.1.3, and from the Shapiro time delay as described in Section 2.1.2.
1. The Mind-Body Problem and the History of Dualism
1.1 The Mind-Body Problem
The mind-body problem is the problem: what is the relationship between mind and body? Or alternatively: what is the relationship between mental properties and physical properties?
Humans have (or seem to have) both physical properties and mental properties. People have (or seem to have)the sort of properties attributed in the physical sciences. These physical properties include size, weight, shape, colour, motion through space and time, etc. But they also have (or seem to have) mental properties, which we do not attribute to typical physical objects These properties involve consciousness (including perceptual experience, emotional experience, and much else), intentionality (including beliefs, desires, and much else), and they are possessed by a subject or a self. Physical properties are public, in the sense that they are, in principle, equally observable by anyone. Some physical properties – like those of an electron – are not directly observable at all, but they are equally available to all, to the same degree, with scientific equipment and techniques. The same is not true of mental properties. I may be able to tell that you are in pain by your behaviour, but only you can feel it directly. Similarly, you just know how something looks to you, and I can only surmise. Conscious mental events are private to the subject, who has a privileged access to them of a kind no-one has to the physical. The mind-body problem concerns the relationship between these two sets of properties. The mind-body problem breaks down into a number of components. The ontological question: what are mental states and what are physical states? Is one class a subclass of the other, so that all mental states are physical, or vice versa? Or are mental states and physical states entirely distinct?
The causal question: do physical states influence mental states? Do mental states influence physical states? If so, how?
Different aspects of the mind-body problem arise for different aspects of the mental, such as consciousness, intentionality, the self. The problem of consciousness: what is consciousness? How is it related to the brain and the body? The problem of intentionality: what is intentionality? How is it related to the brain and the body? The problem of the self: what is the self? How is it related to the brain and the body? Other aspects of the mind-body problem arise for aspects of the physical. For example:
The problem of embodiment: what is it for the mind to be housed in a body? What is it for a body to belong to a particular subject?
The seemingly intractable nature of these problems have given rise to many different philosophical views.
Materialist views say that, despite appearances to the contrary, mental states are just physical states. Behaviourism, functionalism, mind-brain identity theory and the computational theory of mind are examples of how materialists attempt to explain how this can be so. The most common factor in such theories is the attempt to explicate the nature of mind and consciousness in terms of their ability to directly or indirectly modify behaviour, but there are versions of materialism that try to tie the mental to the physical without explicitly explaining the mental in terms of its behaviour-modifying role. The latter are often grouped together under the label ‘non-reductive physicalism’, though this label is itself rendered elusive because of the controversial nature of the term ‘reduction’.
Idealist views say that physical states are really mental. This is because the physical world is an empirical world and, as such, it is the intersubjective product of our collective experience.
Dualist views (the subject of this entry) say that the mental and the physical are both real and neither can be assimilated to the other. For the various forms that dualism can take and the associated problems, see below.
In sum, we can say that there is a mind-body problem because both consciousness and thought, broadly construed, seem very different from anything physical and there is no convincing consensus on how to build a satisfactorily unified picture of creatures possessed of both a mind and a body.
Other entries which concern aspects of the mind-body problem include (among many others): behaviorism, consciousness, eliminative materialism, epiphenomenalism, functionalism, identity theory, intentionality, mental causation, neutral monism, and physicalism.
1.2 History of dualism
In dualism, ‘mind’ is contrasted with ‘body’, but at different times, different aspects of the mind have been the centre of attention. In the classical and mediaeval periods, it was the intellect that was thought to be most obviously resistant to a materialistic account: from Descartes on, the main stumbling block to materialist monism was supposed to be ‘consciousness’, of which phenomenal consciousness or sensation came to be considered as the paradigm instance.
The classical emphasis originates in Plato’s Phaedo. Plato believed that the true substances are not physical bodies, which are ephemeral, but the eternal Forms of which bodies are imperfect copies. These Forms not only make the world possible, they also make it intelligible, because they perform the role of universals, or what Frege called ‘concepts’. It is their connection with intelligibility that is relevant to the philosophy of mind. Because Forms are the grounds of intelligibility, they are what the intellect must grasp in the process of understanding. In Phaedo Plato presents a variety of arguments for the immortality of the soul, but the one that is relevant for our purposes is that the intellect is immaterial because Forms are immaterial and intellect must have an affinity with the Forms it apprehends (78b4–84b8). This affinity is so strong that the soul strives to leave the body in which it is imprisoned and to dwell in the realm of Forms. It may take many reincarnations before this is achieved. Plato’s dualism is not, therefore, simply a doctrine in the philosophy of mind, but an integral part of his whole metaphysics.
One problem with Plato’s dualism was that, though he speaks of the soul as imprisoned in the body, there is no clear account of what binds a particular soul to a particular body. Their difference in nature makes the union a mystery.
Aristotle did not believe in Platonic Forms, existing independently of their instances. Aristotelian forms (the capital ‘F’ has disappeared with their standing as autonomous entities) are the natures and properties of things and exist embodied in those things. This enabled Aristotle to explain the union of body and soul by saying that the soul is the form of the body. This means that a particular person’s soul is no more than his nature as a human being. Because this seems to make the soul into a property of the body, it led many interpreters, both ancient and modern, to interpret his theory as materialistic. The interpretation of Aristotle’s philosophy of mind – and, indeed, of his whole doctrine of form – remains as live an issue today as it was immediately after his death (Robinson 1983 and 1991; Nussbaum 1984; Rorty and Nussbaum, eds, 1992). Nevertheless, the text makes it clear that Aristotle believed that the intellect, though part of the soul, differs from other faculties in not having a bodily organ. His argument for this constitutes a more tightly argued case than Plato’s for the immateriality of thought and, hence, for a kind of dualism. He argued that the intellect must be immaterial because if it were material it could not receive all forms. Just as the eye, because of its particular physical nature, is sensitive to light but not to sound, and the ear to sound and not to light, so, if the intellect were in a physical organ it could be sensitive only to a restricted range of physical things; but this is not the case, for we can think about any kind of material object (De Anima III,4; 429a10–b9). As it does not have a material organ, its activity must be essentially immaterial.
It is common for modern Aristotelians, who otherwise have a high view of Aristotle’s relevance to modern philosophy, to treat this argument as being of purely historical interest, and not essential to Aristotle’s system as a whole. They emphasize that he was not a ‘Cartesian’ dualist, because the intellect is an aspect of the soul and the soul is the form of the body, not a separate substance. Kenny (1989) argues that Aristotle’s theory of mind as form gives him an account similar to Ryle (1949), for it makes the soul equivalent to the dispositions possessed by a living body. This ‘anti-Cartesian’ approach to Aristotle arguably ignores the fact that, for Aristotle, the form is the substance.
These issues might seem to be of purely historical interest. But we shall see in below, in section 4.5, that this is not so.
The identification of form and substance is a feature of Aristotle’s system that Aquinas effectively exploits in this context, identifying soul, intellect and form, and treating them as a substance. (See, for example, Aquinas (1912), Part I, questions 75 and 76.) But though the form (and, hence, the intellect with which it is identical) are the substance of the human person, they are not the person itself. Aquinas says that when one addresses prayers to a saint – other than the Blessed Virgin Mary, who is believed to retain her body in heaven and is, therefore, always a complete person – one should say, not, for example, ‘Saint Peter pray for us’, but ‘soul of Saint Peter pray for us’. The soul, though an immaterial substance, is the person only when united with its body. Without the body, those aspects of its personal memory that depend on images (which are held to be corporeal) will be lost.(See Aquinas (1912), Part I, question 89.)
The more modern versions of dualism have their origin in Descartes’ Meditations, and in the debate that was consequent upon Descartes’ theory. Descartes was a substance dualist. He believed that there were two kinds of substance: matter, of which the essential property is that it is spatially extended; and mind, of which the essential property is that it thinks. Descartes’ conception of the relation between mind and body was quite different from that held in the Aristotelian tradition. For Aristotle, there is no exact science of matter. How matter behaves is essentially affected by the form that is in it. You cannot combine just any matter with any form – you cannot make a knife out of butter, nor a human being out of paper – so the nature of the matter is a necessary condition for the nature of the substance. But the nature of the substance does not follow from the nature of its matter alone: there is no ‘bottom up’ account of substances. Matter is a determinable made determinate by form. This was how Aristotle thought that he was able to explain the connection of soul to body: a particular soul exists as the organizing principle in a particular parcel of matter.
The belief in the relative indeterminacy of matter is one reason for Aristotle’s rejection of atomism. If matter is atomic, then it is already a collection of determinate objects in its own right, and it becomes natural to regard the properties of macroscopic substances as mere summations of the natures of the atoms.
Although, unlike most of his fashionable contemporaries and immediate successors, Descartes was not an atomist, he was, like the others, a mechanist about the properties of matter. Bodies are machines that work according to their own laws. Except where there are minds interfering with it, matter proceeds deterministically, in its own right. Where there are minds requiring to influence bodies, they must work by ‘pulling levers’ in a piece of machinery that already has its own laws of operation. This raises the question of where those ‘levers’ are in the body. Descartes opted for the pineal gland, mainly because it is not duplicated on both sides of the brain, so it is a candidate for having a unique, unifying function.
The main uncertainty that faced Descartes and his contemporaries, however, was not where interaction took place, but how two things so different as thought and extension could interact at all. This would be particularly mysterious if one had an impact view of causal interaction, as would anyone influenced by atomism, for whom the paradigm of causation is like two billiard balls cannoning off one another.
Various of Descartes’ disciples, such as Arnold Geulincx and Nicholas Malebranche, concluded that all mind-body interactions required the direct intervention of God. The appropriate states of mind and body were only the occasions for such intervention, not real causes. Now it would be convenient to think that occasionalists held that all causation was natural except for that between mind and body. In fact they generalized their conclusion and treated all causation as directly dependent on God. Why this was so, we cannot discuss here.
Descartes’ conception of a dualism of substances came under attack from the more radical empiricists, who found it difficult to attach sense to the concept of substance at all. Locke, as a moderate empiricist, accepted that there were both material and immaterial substances. Berkeley famously rejected material substance, because he rejected all existence outside the mind. In his early Notebooks, he toyed with the idea of rejecting immaterial substance, because we could have no idea of it, and reducing the self to a collection of the ‘ideas’ that constituted its contents. Finally, he decided that the self, conceived as something over and above the ideas of which it was aware, was essential for an adequate understanding of the human person. Although the self and its acts are not presented to consciousness as objects of awareness, we are obliquely aware of them simply by dint of being active subjects. Hume rejected such claims, and proclaimed the self to be nothing more than a concatenation of its ephemeral contents.
In fact, Hume criticised the whole conception of substance for lacking in empirical content: when you search for the owner of the properties that make up a substance, you find nothing but further properties. Consequently, the mind is, he claimed, nothing but a ‘bundle’ or ‘heap’ of impressions and ideas – that is, of particular mental states or events, without an owner. This position has been labelled bundle dualism, and it is a special case of a general bundle theory of substance, according to which objects in general are just organised collections of properties. The problem for the Humean is to explain what binds the elements in the bundle together. This is an issue for any kind of substance, but for material bodies the solution seems fairly straightforward: the unity of a physical bundle is constituted by some form of causal interaction between the elements in the bundle. For the mind, mere causal connection is not enough; some further relation of co-consciousness is required. We shall see in 5.2.1 that it is problematic whether one can treat such a relation as more primitive than the notion of belonging to a subject.
One should note the following about Hume’s theory. His bundle theory is a theory about the nature of the unity of the mind. As a theory about this unity, it is not necessarily dualist. Parfit (1970, 1984) and Shoemaker (1984, ch. 2), for example, accept it as physicalists. In general, physicalists will accept it unless they wish to ascribe the unity to the brain or the organism as a whole. Before the bundle theory can be dualist one must accept property dualism, for more about which, see the next section.
A crisis in the history of dualism came, however, with the growing popularity of mechanism in science in the nineteenth century. According to the mechanist, the world is, as it would now be expressed, ‘closed under physics’. This means that everything that happens follows from and is in accord with the laws of physics. There is, therefore, no scope for interference in the physical world by the mind in the way that interactionism seems to require. According to the mechanist, the conscious mind is an epiphenomenon (a notion given general currency by T. H. Huxley 1893): that is, it is a by-product of the physical system which has no influence back on it. In this way, the facts of consciousness are acknowledged but the integrity of physical science is preserved. However, many philosophers found it implausible to claim such things as the following; the pain that I have when you hit me, the visual sensations I have when I see the ferocious lion bearing down on me or the conscious sense of understanding I have when I hear your argument – all have nothing directly to do with the way I respond. It is very largely due to the need to avoid this counterintuitiveness that we owe the concern of twentieth century philosophy to devise a plausible form of materialist monism. But, although dualism has been out of fashion in psychology since the advent of behaviourism (Watson 1913) and in philosophy since Ryle (1949), the argument is by no means over. Some distinguished neurologists, such as Sherrington (1940) and Eccles (Popper and Eccles 1977) have continued to defend dualism as the only theory that can preserve the data of consciousness. Amongst mainstream philosophers, discontent with physicalism led to a modest revival of property dualism in the last decade of the twentieth century. At least some of the reasons for this should become clear below.
2. Varieties of Dualism: Ontology
There are various ways of dividing up kinds of dualism. One natural way is in terms of what sorts of things one chooses to be dualistic about. The most common categories lighted upon for these purposes are substance and property, giving one substance dualism and property dualism. There is, however, an important third category, namely predicate dualism. As this last is the weakest theory, in the sense that it claims least, I shall begin by characterizing it.
2.1 Predicate dualism
Predicate dualism is the theory that psychological or mentalistic predicates are (a) essential for a full description of the world and (b) are not reducible to physicalistic predicates. For a mental predicate to be reducible, there would be bridging laws connecting types of psychological states to types of physical ones in such a way that the use of the mental predicate carried no information that could not be expressed without it. An example of what we believe to be a true type reduction outside psychology is the case of water, where water is always H2O: something is water if and only if it is H2O. If one were to replace the word ‘water’ by ‘H2O’, it is plausible to say that one could convey all the same information. But the terms in many of the special sciences (that is, any science except physics itself) are not reducible in this way. Not every hurricane or every infectious disease, let alone every devaluation of the currency or every coup d’etat has the same constitutive structure. These states are defined more by what they do than by their composition or structure. Their names are classified as functional terms rather than natural kind terms. It goes with this that such kinds of state are multiply realizable; that is, they may be constituted by different kinds of physical structures under different circumstances. Because of this, unlike in the case of water and H2O, one could not replace these terms by some more basic physical description and still convey the same information. There is no particular description, using the language of physics or chemistry, that would do the work of the word ‘hurricane’, in the way that ‘H2O’ would do the work of ‘water’. It is widely agreed that many, if not all, psychological states are similarly irreducible, and so psychological predicates are not reducible to physical descriptions and one has predicate dualism. (The classic source for irreducibility in the special sciences in general is Fodor (1974), and for irreducibility in the philosophy of mind, Davidson (1971).)
2.2 Property Dualism
Whereas predicate dualism says that there are two essentially different kinds of predicates in our language, property dualism says that there are two essentially different kinds of property out in the world. Property dualism can be seen as a step stronger than predicate dualism. Although the predicate ‘hurricane’ is not equivalent to any single description using the language of physics, we believe that each individual hurricane is nothing but a collection of physical atoms behaving in a certain way: one need have no more than the physical atoms, with their normal physical properties, following normal physical laws, for there to be a hurricane. One might say that we need more than the language of physics to describe and explain the weather, but we do not need more than its ontology. There is token identity between each individual hurricane and a mass of atoms, even if there is no type identity between hurricanes as kinds and some particular structure of atoms as a kind. Genuine property dualism occurs when, even at the individual level, the ontology of physics is not sufficient to constitute what is there. The irreducible language is not just another way of describing what there is, it requires that there be something more there than was allowed for in the initial ontology. Until the early part of the twentieth century, it was common to think that biological phenomena (‘life’) required property dualism (an irreducible ‘vital force’), but nowadays the special physical sciences other than psychology are generally thought to involve only predicate dualism. In the case of mind, property dualism is defended by those who argue that the qualitative nature of consciousness is not merely another way of categorizing states of the brain or of behaviour, but a genuinely emergent phenomenon.
2.3 Substance Dualism
There are two important concepts deployed in this notion. One is that of substance, the other is the dualism of these substances. A substance is characterized by its properties, but, according to those who believe in substances, it is more than the collection of the properties it possesses, it is the thing which possesses them. So the mind is not just a collection of thoughts, but is that which thinks, an immaterial substance over and above its immaterial states. Properties are the properties of objects. If one is a property dualist, one may wonder what kinds of objects possess the irreducible or immaterial properties in which one believes. One can use a neutral expression and attribute them to persons, but, until one has an account of person, this is not explanatory. One might attribute them to human beings qua animals, or to the brains of these animals. Then one will be holding that these immaterial properties are possessed by what is otherwise a purely material thing. But one may also think that not only mental states are immaterial, but that the subject that possesses them must also be immaterial. Then one will be a dualist about that to which mental states and properties belong as well about the properties themselves. Now one might try to think of these subjects as just bundles of the immaterial states. This is Hume’s view. But if one thinks that the owner of these states is something quite over and above the states themselves, and is immaterial, as they are, one will be a substance dualist.
Substance dualism is also often dubbed ‘Cartesian dualism’, but some substance dualists are keen to distinguish their theories from Descartes’s. E. J. Lowe, for example, is a substance dualist, in the following sense. He holds that a normal human being involves two substances, one a body and the other a person. The latter is not, however, a purely mental substance that can be defined in terms of thought or consciousness alone, as Descartes claimed. But persons and their bodies have different identity conditions and are both substances, so there are two substances essentially involved in a human being, hence this is a form of substance dualism. Lowe (2006) claims that his theory is close to P. F. Strawson’s (1959), whilst admitting that Strawson would not have called it substance dualism.
3. Varieties of Dualism: Interaction
If mind and body are different realms, in the way required by either property or substance dualism, then there arises the question of how they are related. Common sense tells us that they interact: thoughts and feelings are at least sometimes caused by bodily events and at least sometimes themselves give rise to bodily responses. I shall now consider briefly the problems for interactionism, and its main rivals, epiphenomenalism and parallelism.
3.1 Interactionism
Interactionism is the view that mind and body – or mental events and physical events – causally influence each other. That this is so is one of our common-sense beliefs, because it appears to be a feature of everyday experience. The physical world influences my experience through my senses, and I often react behaviourally to those experiences. My thinking, too, influences my speech and my actions. There is, therefore, a massive natural prejudice in favour of interactionism. It has been claimed, however, that it faces serious problems (some of which were anticipated in section 1).
The simplest objection to interaction is that, in so far as mental properties, states or substances are of radically different kinds from each other, they lack that communality necessary for interaction. It is generally agreed that, in its most naive form, this objection to interactionism rests on a ‘billiard ball’ picture of causation: if all causation is by impact, how can the material and the immaterial impact upon each other? But if causation is either by a more ethereal force or energy or only a matter of constant conjunction, there would appear to be no problem in principle with the idea of interaction of mind and body.
Even if there is no objection in principle, there appears to be a conflict between interactionism and some basic principles of physical science. For example, if causal power was flowing in and out of the physical system, energy would not be conserved, and the conservation of energy is a fundamental scientific law. Various responses have been made to this. One suggestion is that it might be possible for mind to influence the distribution of energy, without altering its quantity. (See Averill and Keating 1981). Another response is to challenge the relevance of the conservation principle in this context. The conservation principle states that ‘in a causally isolated system the total amount of energy will remain constant’. Whereas ‘[t]he interactionist denies…that the human body is an isolated system’, so the principle is irrelevant (Larmer (1986), 282: this article presents a good brief survey of the options). This approach has been termed conditionality, namely the view that conservation is conditional on the physical system being closed, that is, that nothing non-physical is interacting or interfering with it, and, of course, the interactionist claims that this condition is, trivially, not met. That conditionality is the best line for the dualist to take, and that other approaches do not work, is defended in Pitts (2019) and Cucu and Pitts (2019). This, they claim, makes the plausibility of interactionism an empirical matter which only close investigation on the fine operation of the brain could hope to settle. Cucu, in a separate article (2018), claims to find critical neuronal events which do not have sufficient physical explanation.This claim clearly needs further investigation.
Robins Collins (2011) has claimed that the appeal to conservation by opponents of interactionism is something of a red herring because conservation principles are not ubiquitous in physics. He argues that energy is not conserved in general relativity, in quantum theory, or in the universe taken as a whole. Why then, should we insist on it in mind-brain interaction?
Most discussion of interactionism takes place in the context of the assumption that it is incompatible with the world’s being ‘closed under physics’. This is a very natural assumption, but it is not justified if causal overdetermination of behaviour is possible. There could then be a complete physical cause of behaviour, and a mental one. The strongest intuitive objection against overdetermination is clearly stated by Mills (1996: 112), who is himself a defender of overdetermination.
For X to be a cause of Y, X must contribute something to Y. The only way a purely mental event could contribute to a purely physical one would be to contribute some feature not already determined by a purely physical event. But if physical closure is true, there is no feature of the purely physical effect that is not contributed by the purely physical cause. Hence interactionism violates physical closure after all.
Mills says that this argument is invalid, because a physical event can have features not explained by the event which is its sufficient cause. For example, “the rock’s hitting the window is causally sufficient for the window’s breaking, and the window’s breaking has the feature of being the third window-breaking in the house this year; but the facts about prior window-breakings, rather than the rock’s hitting the window, are what cause this window-breaking to have this feature.”
The opponent of overdetermination could perhaps reply that his principle applies, not to every feature of events, but to a subgroup – say, intrinsic features, not merely relational or comparative ones. It is this kind of feature that the mental event would have to cause, but physical closure leaves no room for this. These matters are still controversial.
The problem with closure of physics may be radically altered if physical laws are indeterministic, as quantum theory seems to assert. If physical laws are deterministic, then any interference from outside would lead to a breach of those laws. But if they are indeterministic, might not interference produce a result that has a probability greater than zero, and so be consistent with the laws? This way, one might have interaction yet preserve a kind of nomological closure, in the sense that no laws are infringed. Because it involves assessing the significance and consequences of quantum theory, this is a difficult matter for the non-physicist to assess. Some argue that indeterminacy manifests itself only on the subatomic level, being cancelled out by the time one reaches even very tiny macroscopic objects: and human behaviour is a macroscopic phenomenon. Others argue that the structure of the brain is so finely tuned that minute variations could have macroscopic effects, rather in the way that, according to ‘chaos theory’, the flapping of a butterfly’s wings in China might affect the weather in New York. (For discussion of this, see Eccles (1980), (1987), and Popper and Eccles (1977).) Still others argue that quantum indeterminacy manifests itself directly at a high level, when acts of observation collapse the wave function, suggesting that the mind may play a direct role in affecting the state of the world (Hodgson 1988; Stapp 1993).
3.2 Epiphenomenalism
If the reality of property dualism is not to be denied, but the problem of how the immaterial is to affect the material is to be avoided, then epiphenomenalism may seem to be the answer. According to this theory, mental events are caused by physical events, but have no causal influence on the physical. I have introduced this theory as if its point were to avoid the problem of how two different categories of thing might interact. In fact, it is, at best, an incomplete solution to this problem. If it is mysterious how the non-physical can have it in its nature to influence the physical, it ought to be equally mysterious how the physical can have it in its nature to produce something non-physical. But that this latter is what occurs is an essential claim of epiphenomenalism. (For development of this point, see Green (2003), 149–51). In fact, epiphenomenalism is more effective as a way of saving the autonomy of the physical (the world as ‘closed under physics’) than as a contribution to avoiding the need for the physical and non-physical to have causal commerce.
There are at least three serious problems for epiphenomenalism. First, as I indicated in section 1, it is profoundly counterintuitive. What could be more apparent than that it is the pain that I feel that makes me cry, or the visual experience of the boulder rolling towards me that makes me run away? At least one can say that epiphenomenalism is a fall-back position: it tends to be adopted because other options are held to be unacceptable.
The second problem is that, if mental states do nothing, there is no reason why they should have evolved. This objection ties in with the first: the intuition there was that conscious states clearly modify our behaviour in certain ways, such as avoiding danger, and it is plain that they are very useful from an evolutionary perspective.
Frank Jackson (1982) replies to this objection by saying that it is the brain state associated with pain that evolves for this reason: the sensation is a by-product. Evolution is full of useless or even harmful by-products. For example, polar bears have evolved thick coats to keep them warm, even though this has the damaging side effect that they are heavy to carry. Jackson’s point is true in general, but does not seem to apply very happily to the case of mind. The heaviness of the polar bear’s coat follows directly from those properties and laws which make it warm: one could not, in any simple way, have one without the other. But with mental states, dualistically conceived, the situation is quite the opposite. The laws of physical nature which, the mechanist says, make brain states cause behaviour, in no way explain why brain states should give rise to conscious ones. The laws linking mind and brain are what Feigl (1958) calls nomological danglers, that is, brute facts added onto the body of integrated physical law. Why there should have been by-products of that kind seems to have no evolutionary explanation.
The third problem concerns the rationality of belief in epiphenomenalism, via its effect on the problem of other minds. It is natural to say that I know that I have mental states because I experience them directly. But how can I justify my belief that others have them? The simple version of the ‘argument from analogy’ says that I can extrapolate from my own case. I know that certain of my mental states are correlated with certain pieces of behaviour, and so I infer that similar behaviour in others is also accompanied by similar mental states. Many hold that this is a weak argument because it is induction from one instance, namely, my own. The argument is stronger if it is not a simple induction but an ‘argument to the best explanation’. I seem to know from my own case that mental events can be the explanation of behaviour, and I know of no other candidate explanation for typical human behaviour, so I postulate the same explanation for the behaviour of others. But if epiphenomenalism is true, my mental states do not explain my behaviour and there is a physical explanation for the behaviour of others. It is explanatorily redundant to postulate such states for others. I know, by introspection, that I have them, but is it not just as likely that I alone am subject to this quirk of nature, rather than that everyone is?
For more detailed treatment and further reading on this topic, see the entry epiphenomenalism.
3.3 Parallelism
The epiphenomenalist wishes to preserve the integrity of physical science and the physical world, and appends those mental features that he cannot reduce. The parallelist preserves both realms intact, but denies all causal interaction between them. They run in harmony with each other, but not because their mutual influence keeps each other in line. That they should behave as if they were interacting would seem to be a bizarre coincidence. This is why parallelism has tended to be adopted only by those – like Leibniz – who believe in a pre-established harmony, set in place by God. The progression of thought can be seen as follows. Descartes believes in a more or less natural form of interaction between immaterial mind and material body. Malebranche thought that this was impossible naturally, and so required God to intervene specifically on each occasion on which interaction was required. Leibniz decided that God might as well set things up so that they always behaved as if they were interacting, without particular intervention being required. Outside such a theistic framework, the theory is incredible. Even within such a framework, one might well sympathise with Berkeley’s instinct that once genuine interaction is ruled out one is best advised to allow that God creates the physical world directly, within the mental realm itself, as a construct out of experience.
4. Arguments for Dualism
4.1 The Knowledge Argument Against Physicalism
One category of arguments for dualism is constituted by the standard objections against physicalism. Prime examples are those based on the existence of qualia, the most important of which is the so-called ‘knowledge argument’. Because this argument has its own entry (see the entry qualia: the knowledge argument), I shall deal relatively briefly with it here. One should bear in mind, however, that all arguments against physicalism are also arguments for the irreducible and hence immaterial nature of the mind and, given the existence of the material world, are thus arguments for dualism.
The knowledge argument asks us to imagine a future scientist who has lacked a certain sensory modality from birth, but who has acquired a perfect scientific understanding of how this modality operates in others. This scientist – call him Harpo – may have been born stone deaf, but become the world’s greatest expert on the machinery of hearing: he knows everything that there is to know within the range of the physical and behavioural sciences about hearing. Suppose that Harpo, thanks to developments in neurosurgery, has an operation which finally enables him to hear. It is suggested that he will then learn something he did not know before, which can be expressed as what it is like to hear, or the qualitative or phenomenal nature of sound. These qualitative features of experience are generally referred to as qualia. If Harpo learns something new, he did not know everything before. He knew all the physical facts before. So what he learns on coming to hear – the facts about the nature of experience or the nature of qualia – are non-physical. This establishes at least a state or property dualism. (See Jackson 1982; Robinson 1982.)
There are at least two lines of response to this popular but controversial argument. First is the ‘ability’ response. According to this, Harpo does not acquire any new factual knowledge, only ‘knowledge how’, in the form of the ability to respond directly to sounds, which he could not do before. This essentially behaviouristic account is exactly what the intuition behind the argument is meant to overthrow. Putting ourselves in Harpo’s position, it is meant to be obvious that what he acquires is knowledge of what something is like, not just how to do something. Such appeals to intuition are always, of course, open to denial by those who claim not to share the intuition. Some ability theorists seem to blur the distinction between knowing what something is like and knowing how to do something, by saying that the ability Harpo acquires is to imagine or remember the nature of sound. In this case, what he acquires the ability to do involves the representation to himself of what the thing is like. But this conception of representing to oneself, especially in the form of imagination, seems sufficiently close to producing in oneself something very like a sensory experience that it only defers the problem: until one has a physicalist gloss on what constitutes such representations as those involved in conscious memory and imagination, no progress has been made.
The other line of response is to argue that, although Harpo’s new knowledge is factual, it is not knowledge of a new fact. Rather, it is new way of grasping something that he already knew. He does not realise this, because the concepts employed to capture experience (such as ‘looks red’ or ‘sounds C-sharp’) are similar to demonstratives, and demonstrative concepts lack the kind of descriptive content that allow one to infer what they express from other pieces of information that one may already possess. A total scientific knowledge of the world would not enable you to say which time was ‘now’ or which place was ‘here’. Demonstrative concepts pick something out without saying anything extra about it. Similarly, the scientific knowledge that Harpo originally possessed did not enable him to anticipate what it would be like to re-express some parts of that knowledge using the demonstrative concepts that only experience can give one. The knowledge, therefore, appears to be genuinely new, whereas only the mode of conceiving it is novel.
Proponents of the epistemic argument respond that it is problematic to maintain both that the qualitative nature of experience can be genuinely novel, and that the quality itself be the same as some property already grasped scientifically: does not the experience’s phenomenal nature, which the demonstrative concepts capture, constitute a property in its own right? Another way to put this is to say that phenomenal concepts are not pure demonstratives, like ‘here’ and ‘now’, or ‘this’ and ‘that’, because they do capture a genuine qualitative content. Furthermore, experiencing does not seem to consist simply in exercising a particular kind of concept, demonstrative or not. When Harpo has his new form of experience, he does not simply exercise a new concept; he also grasps something new – the phenomenal quality – with that concept. How decisive these considerations are, remains controversial.
4.2 The Argument from Predicate Dualism to Property Dualism
I said above that predicate dualism might seem to have no ontological consequences, because it is concerned only with the different way things can be described within the contexts of the different sciences, not with any real difference in the things themselves. This, however, can be disputed.
The argument from predicate to property dualism moves in two steps, both controversial. The first claims that the irreducible special sciences, which are the sources of irreducible predicates, are not wholly objective in the way that physics is, but depend for their subject matter upon interest-relative perspectives on the world. This means that they, and the predicates special to them, depend on the existence of minds and mental states, for only minds have interest-relative perspectives. The second claim is that psychology – the science of the mental – is itself an irreducible special science, and so it, too, presupposes the existence of the mental. Mental predicates therefore presuppose the mentality that creates them: mentality cannot consist simply in the applicability of the predicates themselves.
First, let us consider the claim that the special sciences are not fully objective, but are interest-relative.
No-one would deny, of course, that the very same subject matter or ‘hunk of reality’ can be described in irreducibly different ways and it still be just that subject matter or piece of reality. A mass of matter could be characterized as a hurricane, or as a collection of chemical elements, or as mass of sub-atomic particles, and there be only the one mass of matter. But such different explanatory frameworks seem to presuppose different perspectives on that subject matter.
This is where basic physics, and perhaps those sciences reducible to basic physics, differ from irreducible special sciences. On a realist construal, the completed physics cuts physical reality up at its ultimate joints: any special science which is nomically strictly reducible to physics also, in virtue of this reduction, it could be argued, cuts reality at its joints, but not at its minutest ones. If scientific realism is true, a completed physics will tell one how the world is, independently of any special interest or concern: it is just how the world is. It would seem that, by contrast, a science which is not nomically reducible to physics does not take its legitimation from the underlying reality in this direct way. Rather, such a science is formed from the collaboration between, on the one hand, objective similarities in the world and, on the other, perspectives and interests of those who devise the science. The concept of hurricane is brought to bear from the perspective of creatures concerned about the weather. Creatures totally indifferent to the weather would have no reason to take the real patterns of phenomena that hurricanes share as constituting a single kind of thing. With the irreducible special sciences, there is an issue of salience , which involves a subjective component: a selection of phenomena with a certain teleology in mind is required before their structures or patterns are reified. The entities of metereology or biology are, in this respect, rather like Gestalt phenomena.
Even accepting this, why might it be thought that the perspectivality of the special sciences leads to a genuine property dualism in the philosophy of mind? It might seem to do so for the following reason. Having a perspective on the world, perceptual or intellectual, is a psychological state. So the irreducible special sciences presuppose the existence of mind. If one is to avoid an ontological dualism, the mind that has this perspective must be part of the physical reality on which it has its perspective. But psychology, it seems to be almost universally agreed, is one of those special sciences that is not reducible to physics, so if its subject matter is to be physical, it itself presupposes a perspective and, hence, the existence of a mind to see matter as psychological. If this mind is physical and irreducible, it presupposes mind to see it as such. We seem to be in a vicious circle or regress.
We can now understand the motivation for full-blown reduction. A true basic physics represents the world as it is in itself, and if the special sciences were reducible, then the existence of their ontologies would make sense as expressions of the physical, not just as ways of seeing or interpreting it. They could be understood ‘from the bottom up’, not from top down. The irreducibility of the special sciences creates no problem for the dualist, who sees the explanatory endeavor of the physical sciences as something carried on from a perspective conceptually outside of the physical world. Nor need this worry a physicalist, if he can reduce psychology, for then he could understand ‘from the bottom up’ the acts (with their internal, intentional contents) which created the irreducible ontologies of the other sciences. But psychology is one of the least likely of sciences to be reduced. If psychology cannot be reduced, this line of reasoning leads to real emergence for mental acts and hence to a real dualism for the properties those acts instantiate (Robinson 2003).
4.3 The Modal Argument
There is an argument, which has roots in Descartes (Meditation VI), which is a modal argument for dualism. One might put it as follows:
It is imaginable that one’s mind might exist without one’s body.
therefore
It is conceivable that one’s mind might exist without one’s body.
therefore
It is possible one’s mind might exist without one’s body.
therefore
One’s mind is a different entity from one’s body.
The rationale of the argument is a move from imaginability to real possibility. I include (2) because the notion of conceivability has one foot in the psychological camp, like imaginability, and one in the camp of pure logical possibility and therefore helps in the transition from one to the other.
This argument should be distinguished from a similar ‘conceivability’ argument, often known as the ‘zombie hypothesis’, which claims the imaginability and possibility of my body (or, in some forms, a body physically just like it) existing without there being any conscious states associated with it. (See, for example, Chalmers (1996), 94–9.) This latter argument, if sound, would show that conscious states were something over and above physical states. It is a different argument because the hypothesis that the unaltered body could exist without the mind is not the same as the suggestion that the mind might continue to exist without the body, nor are they trivially equivalent. The zombie argument establishes only property dualism and a property dualist might think disembodied existence inconceivable – for example, if he thought the identity of a mind through time depended on its relation to a body (e.g., Penelhum 1970).
Before Kripke (1972/80), the first challenge to such an argument would have concerned the move from (3) to (4). When philosophers generally believed in contingent identity, that move seemed to them invalid. But nowadays that inference is generally accepted and the issue concerns the relation between imaginability and possibility. No-one would nowadays identify the two (except, perhaps, for certain quasi-realists and anti-realists), but the view that imaginability is a solid test for possibility has been strongly defended. W. D. Hart ((1994), 266), for example, argues that no clear example has been produced such that “one can imagine that p (and tell less imaginative folk a story that enables them to imagine that p) plus a good argument that it is impossible that p. No such counterexamples have been forthcoming…” This claim is at least contentious. There seem to be good arguments that time-travel is incoherent, but every episode of Star-Trek or Doctor Who shows how one can imagine what it might be like were it possible.
It is worth relating the appeal to possibility in this argument to that involved in the more modest, anti-physicalist, zombie argument. The possibility of this hypothesis is also challenged, but all that is necessary for a zombie to be possible is that all and only the things that the physical sciences say about the body be true of such a creature. As the concepts involved in such sciences – e.g., neuron, cell, muscle – seem to make no reference, explicit or implicit, to their association with consciousness, and are defined in purely physical terms in the relevant science texts, there is a very powerful prima facie case for thinking that something could meet the condition of being just like them and lack any connection with consciousness. There is no parallel clear, uncontroversial and regimented account of mental concepts as a whole that fails to invoke, explicitly or implicitly, physical (e.g., behavioural) states.
For an analytical behaviourist the appeal to imaginability made in the argument fails, not because imagination is not a reliable guide to possibility, but because we cannot imagine such a thing, as it is a priori impossible. The impossibility of disembodiment is rather like that of time travel, because it is demonstrable a priori, though only by arguments that are controversial. The argument can only get under way for those philosophers who accept that the issue cannot be settled a priori, so the possibility of the disembodiment that we can imagine is still prima facie open.
A major rationale of those who think that imagination is not a safe indication of possibility, even when such possibility is not eliminable a priori, is that we can imagine that a posteriori necessities might be false – for example, that Hesperus might not be identical to Phosphorus. But if Kripke is correct, that is not a real possibility. Another way of putting this point is that there are many epistemic possibilities which are imaginable because they are epistemic possibilities, but which are not real possibilities. Richard Swinburne (1997, New Appendix C), whilst accepting this argument in general, has interesting reasons for thinking that it cannot apply in the mind-body case. He argues that in cases that involve a posteriori necessities, such as those identities that need discovering, it is because we identify those entities only by their ‘stereotypes’ (that is, by their superficial features observable by the layman) that we can be wrong about their essences. In the case of our experience of ourselves this is not true.
Now it is true that the essence of Hesperus cannot be discovered by a mere thought experiment. That is because what makes Hesperus Hesperus is not the stereotype, but what underlies it. But it does not follow that no one can ever have access to the essence of a substance, but must always rely for identification on a fallible stereotype. One might think that for the person him or herself, while what makes that person that person underlies what is observable to others, it does not underlie what is experienceable by that person, but is given directly in their own self-awareness.
This is a very appealing Cartesian intuition: my identity as the thinking thing that I am is revealed to me in consciousness, it is not something beyond the veil of consciousness. Now it could be replied to this that though I do access myself as a conscious subject, so classifying myself is rather like considering myself qua cyclist. Just as I might never have been a cyclist, I might never have been conscious, if things had gone wrong in my very early life. I am the organism, the animal, which might not have developed to the point of consciousness, and that essence as animal is not revealed to me just by introspection.
But there are vital differences between these cases. A cyclist is explicitly presented as a human being (or creature of some other animal species) cycling: there is no temptation to think of a cyclist as a basic kind of thing in its own right. Consciousness is not presented as a property of something, but as the subject itself. Swinburne’s claim that when we refer to ourselves we are referring to something we think we are directly aware of and not to ‘something we know not what’ that underlies our experience seemingly ‘of ourselves’ has powerful intuitive appeal and could only be overthrown by very forceful arguments. Yet, even if we are not referring primarily to a substrate, but to what is revealed in consciousness, could it not still be the case that there is a necessity stronger than causal connecting this consciousness to something physical? To consider this further we must investigate what the limits are of the possible analogy between cases of the water-H2O kind, and the mind-body relation.
We start from the analogy between the water stereotype – how water presents itself – and how consciousness is given first-personally to the subject. It is plausible to claim that something like water could exist without being H2O, but hardly that it could exist without some underlying nature. There is, however, no reason to deny that this underlying nature could be homogenous with its manifest nature: that is, it would seem to be possible that there is a world in which the water-like stuff is an element, as the ancients thought, and is water-like all the way down. The claim of the proponents of the dualist argument is that this latter kind of situation can be known to be true a priori in the case of the mind: that is, one can tell by introspection that it is not more-than-causally dependent on something of a radically different nature, such as a brain or body. What grounds might one have for thinking that one could tell that a priori?
The only general argument that seem to be available for this would be the principle that, for any two levels of discourse, A and B, they are more-than-causally connected only if one entails the other a priori. And the argument for accepting this principle would be that the relatively uncontroversial cases of a posteriori necessary connections are in fact cases in which one can argue a priori from facts about the microstructure to the manifest facts. In the case of water, for example, it would be claimed that it follows a priori that if there were something with the properties attributed to H2O by chemistry on a micro level, then that thing would possess waterish properties on a macro level. What is established a posteriori is that it is in fact H2O that underlies and explains the waterish properties round here, not something else: the sufficiency of the base – were it to obtain – to explain the phenomena, can be deduced a priori from the supposed nature of the base. This is, in effect, the argument that Chalmers uses to defend the zombie hypothesis. The suggestion is that the whole category of a posteriori more-than-causally necessary connections (often identified as a separate category of metaphysical necessity) comes to no more than this. If we accept that this is the correct account of a posteriori necessities, and also deny the analytically reductionist theories that would be necessary for a priori connections between mind and body, as conceived, for example, by the behaviourist or the functionalist, does it follow that we can tell a priori that consciousness is not more-than-causally dependent on the body?
It is helpful in considering this question to employ a distinction like Berkeley’s between ideas and notions. Ideas are the objects of our mental acts, and they capture transparently – ‘by way of image or likeness’ (Principles, sect. 27) – that of which they are the ideas. The self and its faculties are not the objects of our mental acts, but are captured only obliquely in the performance of its acts, and of these Berkeley says we have notions, meaning by this that what we capture of the nature of the dynamic agent does not seem to have the same transparency as what we capture as the normal objects of the agent’s mental acts. It is not necessary to become involved in Berkeley’s metaphysics in general to feel the force of the claim that the contents and internal objects of our mental acts are grasped with a lucidity that exceeds that of our grasp of the agent and the acts per se. Because of this, notions of the self perhaps have a ‘thickness’ and are permanently contestable: there seems always to be room for more dispute as to what is involved in that concept. (Though we shall see later, in 5.2.2, that there is a ‘non-thick’ way of taking the Berkeleyan concept of a notion.)
Because ‘thickness’ always leaves room for dispute, this is one of those cases in philosophy in which one is at the mercy of the arguments philosophers happen to think up. The conceivability argument creates a prima facie case for thinking that mind has no more than causal ontological dependence on the body. Let us assume that one rejects analytical (behaviourist or functionalist) accounts of mental predicates. Then the above arguments show that any necessary dependence of mind on body does not follow the model that applies in other scientific cases. This does not show that there may not be other reasons for believing in such dependence, for so many of the concepts in the area are still contested. For example, it might be argued that identity through time requires the kind of spatial existence that only body can give: or that the causal continuity required by a stream of consciousness cannot be a property of mere phenomena. All these might be put forward as ways of filling out those aspects of our understanding of the self that are only obliquely, not transparently, presented in self-awareness. The dualist must respond to any claim as it arises: the conceivability argument does not pre-empt them.......
5.2 The Unity of the Mind
Whether one believes that the mind is a substance or just a bundle of properties, the same challenge arises, which is to explain the nature of the unity of the immaterial mind. For the Cartesian, that means explaining how he understands the notion of immaterial substance. For the Humean, the issue is to explain the nature of the relationship between the different elements in the bundle that binds them into one thing. Neither tradition has been notably successful in this latter task: indeed, Hume, in the appendix to the Treatise, declared himself wholly mystified by the problem, rejecting his own initial solution (though quite why is not clear from the text).
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Sydney Padua's cartoon effort here seems to be the world's only coherent effort to graphically visualize a full-scale Analytical Engine.
Illustration of a web analytics framework - data gathering, data reporting, data analysis - then the bonus stage of optimisation.
Inspired by a blog post by Avinash Kaushik (Occam's Razor)
www.kaushik.net/avinash/web-analytics-consulting-framewor...
Have you checked out Google Analytics? I have been using it for about two months on my blog, and it is very interesting to track...
Sure, it's not about quantity, but I do like the graphs :)
The Map Overlay may be my favorite of all! how many people visited the site from Sri Lanka? you can find it out with Google Analytics!
Have you checked out Google Analytics? I have been using it for about two months on my blog, and it is very interesting to track...
Sure, it's not about quantity, but I do like the graphs :)
The Jack Welch College of Business and the Office of Alumni Engagement presented “Careers in Analytics” on April 10, 2019, at the Martire Forum. The alumni panel featured Justin Baigert ’05, vice president, Data & Analytics at GE, Joseph Lucibello ’11, senior manager, data scientist at WWE and Suzanne May ’13, research manager at Purchased. The moderator was Khawaja Mamun, associate professor of economics. Photo by Mark F. Conrad