Albert Einstein’s Own Words on His General Theory of Relativity: A Philosophical Commentary

 

This essay will raise largely philosophical issues concerning Albert Einstein’s general theory of relativity. Such issues will be prompted by Einstein’s own words on this matter. The essay will also include those intuitive questions which laypersons may ask about Einstein’s theory. I’ll also assume that at least a certain degree of ignorance (or naivety) — on my own part — will be displayed about the detailed physics of general relativity.

These philosophical questions and issues should be taken within the context of there being a large amount of mathematics and (as it were) pure physics in Einstein’s Relativity: The Special and the General Theory — far more than you’d see in most (or even all ) contemporary “popular science” books. Indeed there are equations and mathematical symbols on virtually every page. And all that despite the fact that Einstein aimed his book at those “who are not conversant with the mathematical apparatus of theoretical physics”.

Relativity: The Special and the General Theory began life as a short paper and it was first published in 1916. (My own English edition dates from 1920; which is the same edition displayed in the image above.)

To quote Einstein’s own preface (part of which has just been quoted), this book’s aim is to give

“an exact insight into the theory of relativity to those readers who, from a general scientific and philosophical point of view, are interested in the theory, but who are not conversant with the mathematical apparatus of theoretical physics”.

The Special Theory of Relativity

Since this essay in on the general theory of relativity and I’ll be using Einstein’s own words as a springboard, then it’s wise to begin by quoting Einstein himself distinguishing his general theory of relativity (1915) from his special theory of relativity (1905).

Einstein makes an important distinction in the following:

“[A]ccording to the general theory of relativity, the law of the constancy of the velocity of light in vacuo, which constitutes one of the two fundamental assumptions in the special theory of relativity [] cannot claim any unlimited validity. A curvature of rays of light can only take place when the velocity of propagation of light varies with position.”

Then Einstein makes it clear that his own (prior) special theory of relativity is not thereby dead or irrelevant. Thus:

"Now we might think that as a consequence of this, the special theory of relativity would be laid in the dust. But in reality this is not the case. We can only conclude that the special theory of relativity cannot claim an unlimited domain of validity; its results hold only so long as we are able to disregard the influences of gravitational fields on the phenomena (e.g. of light.)”

This means that it is gravity that’s the vital addition to Relativity when it comes to the general theory. And when gravity and its effects were introduced, then a lot more of physics and cosmology needed to be modified — or even drastically changed — too.

The General Theory of Relativity

Matter and Space

It’s now well known that Albert Einstein argued that space and time are intimately connected — hence the introduction of the notion spacetime. However, he argued that space and matter are intimately connected too. Of course these two separate unifications (i.e., space and time & matter and space) are themselves placed in a larger unification in Einstein’s overall scheme.

Einstein believed that space is “determined by matter”. More technically, he believed that it’s the “geometrical properties of space” which are determined by matter. Thus when matter determines (this word is, so far, vague) a given area of space, then that space’s geometrical properties change.

Einstein concluded with the following words:

“Thus we can draw conclusions about the geometrical structure of the universe only if we base our considerations on the state of the matter as being something that is known.”

This is basically to say that in order to know about the geometrical structure of a given area of space (or indeed the universe as a whole), then we also need to know about “the state of the matter” in that given area of space. Thus if we know about that matter, then we also (or can also) know about that spatial area’s geometrical properties.

So does that also work the other way around?

That is, in order to know about a given piece of matter, do we also need to know about the geometrical structure of the area of space in which that piece of matter is (as it were) imbedded?

All the above raises a question similar to the one asked about the precise relation between mind and matter/the brain (i.e., if we accept some form of dualism). Basically, if the brain and mind are completely different (to use a philosophical term) substances, then how do they interact at all? Similarly, if space and matter are completely different substances(?), then how do they interact at all?

Again, Einstein told us that “the geometrical properties of space are not independent”: they are “determined by matter”. Does that mean that — as with the physicalist notion of mind — space is a kind of matter? Perhaps the least we can say is that space is not nothing. That is, we don’t need to say that space is matter — simply that it is in some way… physical. Indeed if space isn’t in some way physical, then how could it be determined by matter?

So how, exactly, does matter “determine” space’s geometrical properties?

Not only that: we still don’t really know what space and matter are. Perhaps if we can know what they are, then we’d also come to know how matter determines the geometrical properties of space.

Gravitational Fields and Distributions of Matter

Einstein gave us a concrete example of the reciprocal relation between gravitational fields and matter. He wrote:

“We already know from our previous discussion that the behaviour of measuring-rods and clocks is influenced by gravitational fields, i.e. by the distribution of matter.”

Firstly, it must be said that Einstein seemed to be setting up an identity relation here. Thus:

gravitational fields = the distribution of matter

So is it that gravitational fields are an effect of distributions of matter or that they actually are distributions of matter? In any case, the least we can say (at this juncture) is that there’s an intimate relation between gravitational fields and distributions of matter.

This raises the following question: What is a gravitational field?

In any case, even if gravitational fields couldn’t exist without distributions of matter, that doesn’t also mean that they are one and the same thing. In addition, it may still be the case that both gravitational fields and distributions of matter are physical; though not physical in precisely the same way.

Indeed if we accept Einstein’s general theory of relativity, then there’s a way of looking at these issues in which there’s no gravitational force at all. More technically, gravitational fields simply “represent” the curvature of spacetime. This means that gravity is a “fictitious force”. Thus we don’t really need to ask the question, “What is gravity?” And that’s because gravity doesn’t actually exist. (These “interpretations” may not matter to many physicists in that they don’t alter the data.)

We’re still left with this question: “What is spacetime?”

Rigid Bodies?

Einstein often referred to “rigid bodies” in his Relativity: The Special and the General Theory.

Rigid bodies can be characterised negatively in this way:

Rigid bodies are bodies which are not influenced by gravitational fields.

Basically, a rigid body is a body that retains all its geometrical over time.

Yet since gravitational fields exist, then there are no such things as rigid bodies. A non-rigid (Einsteinian) body, on the other hand, is constantly influenced by gravitational fields. And that’s because such fields never (as it were) go away.

More concretely, Einstein told us that a “rigid rod” is not actually rigid at all. He wrote:

“This proves that the propositions of Euclidean geometry cannot hold exactly on the rotating disc, nor in general in a gravitational field, at least if we attribute the length I to the rod in all positions and in every orientation.”

So if a rod is length I at one place and at one time, then it won’t still be length I at another place and at another time. (It must be borne in mind that the differences here are fantastically small!) Yet at that other place and other time it’s still exactly the same rod! To repeat: every change of position and every change of orientation of this rod (which was initially designated as having length I) will change its geometry (or, more simply, its size).

(Philosophers have had fun with the “standard rods”, etc. which are used for measurement — see ‘Kripke, Duchamp & the Standard Metre’.)

Clocks

Einstein then stated that gravitational fields (or various distributions of matter) “influence [] the behaviour of measuring-rods and clocks”. So here again we can ask if it’s the case that physical things (i.e., gravitational fields) are influencing other physical things (i.e., measuring-rods and clocks).

Einstein supplies detail about gravitational effects on clocks. (In this case, it’s in relation to a rotating circular disk on which two clocks are placed.) He wrote:

“Thus on our circular [rotating] disc, to make the case more general, in every gravitational field, a clock will go more quickly or less quickly, according to the position in which the clock is situated (at rest).”

In other words, a clock will either go “more quickly or less quickly” depending on its precise physical and spatial relation to the gravitational field. This means that there is no absolute time which can be supplied by the clock: the time it gives is dependent on its relation to (or place within) a gravitational field. As Einstein put it:

“For this reason it is not possible to obtain a reasonable definition of time with the aid of clocks which are arranged at rest with respect to the body of reference.”

So even if these two clocks were set at the same time and given the same time, then, after those settings, differences will occur which have nothing to do with the clocks’ respective mechanisms and everything to do with their relative positions in the gravitational field. That also means that there’s no absolute way of choosing which of the two clocks (in two different places) is giving the right time. They both are. And that’s the case even though they give — very slightly — different times.

It’s not only the relative positions of the clocks which determines their different times: it’s also a question of their velocities. Einstein wrote:

“As judged from this body, the clock at the centre of the disc has no velocity, whereas the clock at the edge of the disc is in motion relative to K in consequence of the rotation [].”

Thus one clock has no velocity (relative to K) and the other clock is in motion (relative to K). What follows from that? According to Einstein,

“it follows that the latter clock goes at a rate permanently slower than that of the clock at the centre of the circular disc, i.e. as observed from K”.

The clock in motion, then, is slower precisely because it’s in motion. Similarly, the clock at the centre of the disk (which has no velocity) is running more quickly. In other words, velocity (roughly, speed with direction) slows time down. Or, at least at this juncture, velocity slows clock time down.

So, in the case of the two clocks in Einstein’s thought experiment, we can ask this question:

Is it that gravitational fields makes one clock run slower or is it that time itself — for that clock — runs slower?

In other words, is there a difference between clock time and time itself?

The problem here is that if we have no other way of measuring time (or knowing about time) other than via clocks or other (moving) “bodies”, then surely that which measures time and time itself are intimately connected — at least in Einstein’s picture.

In Einstein’s own words, “a physical definition of time” depends on both gravitational fields and how those fields influence clocks. Indeed this fact, as Einstein concedes, wasn’t factored into his special theory of relativity. In other words, time-telling (though not time itself) is relative in the general theory of relativity; whereas in the special theory of relativity it isn’t. That is, although the relativity of time was obviously recognised in the special theory of relativity, the (as it were) relativity of clocks (or other time-telling bodies) wasn’t.

We now also need to ask how, exactly, gravitational fields (to use Einstein’s vague non-technical word) “influence” rigid bodies such as clocks and measuring-rods.

Just a final note on Einstein’s focus on clocks, measuring-rods and rigid bodies (as also featured in the next section).

At this stage of his career, Einstein didn’t attempt to tell his readers what space and time actually are — despite my own questions so far. Instead he took an operationalist (though that term came later) position on these matters. Indeed none other than Alan Turing (1912–1954) picked up on this after reading Einstein. He expressed Einstein’s (1916) position (as quoted by Andrew Hodges) in the following passage:

“It is meaningless to ask whether the two p[oin]ts are always the same distance apart, as you stipulate that the distance is your unit and your ideas have to go by that definition… These ways of measuring are really conventions. You modify your laws to suit your method of measurement.”

Of course Turing himself applied this line of reasoning to this question: “Can a machine think?” In other words, he answered that question operationally in terms of what can be shown to be an example of thinking under controlled circumstances.

Let’s now move on to physical space.

Physical Space

The (as it were) physicality of space has been directly known since Faraday and Maxwell and — in a sense — indirectly known since Newton. The following is how Einstein put it:

“The success of the Faraday-Maxwell interpretation of electromagnetic action at a distance resulted in physicists becoming convinced that there are no such things as instantaneous actions at a distance (not involving an intermediary medium) of the type of Newton’s law of gravitation.”

More particularly, the physicality of space was highlighted when Einstein asked us to “imagine a spherical space”.

Was this a case of Einstein asking us to imagine (a) space without something in it? Or to put that another way: Was Einstein asking us to imagine space itself being spherical?

These questions are asked because in order to imagine a spherical space, Einstein actually imported rigid bodies — again!— into his act of imagination. To use Einstein’s own words:

“To imagine a space means nothing else that that we imagine an epitome of our ‘space’ experience, i.e. of experience that we can have in the movement of ‘rigid’ bodies. In this sense we can imagine a spherical space.”

This basically means that we actually imagine “the movement of rigid bodies” within a given space. And from that “experience” we can conceive (rather than imagine) of space itself being spherical. That is, a clue to the sphericity of space is provided by the movement (or trajectories) of rigid bodies. That means that the imagination — or even observation — of space alone can’t tell us that space is spherical. Instead, it’s the movement of bodies within space which must show us that this is so.

Again: Einstein wasn’t asking us to imagine space with nothing in it. (Perhaps that would be an impossible act of imagination.) Instead we must imagine moving bodies within space and from that act of imagination deduce (or “conceive” — a word which Einstein used later) that space is indeed spherical.

Thus it’s only in (to use Einstein’s own words) “this sense” that we can imagine a spherical space.

In addition, we can only make sense of space when we — in some cases at least — bring in “motion relative to a practically rigid body of reference”. Thus space still exists; though we can only make sense of it when we include a dynamics which is itself a product of introducing rigid bodies and frames of reference.

Another demonstration of the physicality and shape of space is offered by Einstein in his following words:

“At first, the straight lines which radiate from the starting point diverge farther and farther from one another, but later they approach each other, and finally they run together again at a ‘counter-point’ to the starting point. Under such conditions they have traversed the whole spherical space.”

In Euclidean (flat) space, these straight lines would keep on diverging farther and farther from each other. Indeed there would be nothing to stop this from happening. But because (Einstein’s) space is spherical (or curved), then at some point the two lines must eventually “approach each other, and finally [] run together [] at a ‘counter-point’ to the starting point”. In other words, the two lines have journeyed around a spherical space. So this is roughly equivalent to two lines being drawn from a given point on the surface of (say) a football, the two lines then going in different (or opposite) directions, and then the lines finding their way back to a “counter-point to the starting point”.

Yet perhaps the best known of Einstein’s examples of the physicality of space was actually cosmological in extent. This is Einstein’s theory of the bending of light. Einstein himself wrote:

“If the displacement of spectral lines toward the red by the gravitational potential does not exist, then the general theory of relativity will be untenable.”

To state the obvious: if space weren’t physical, then these spectral lines wouldn’t become bent or displaced. Thus the space through which these spectral lines traverse has a geometry which determines their movement. And that movement is not in a straight line.

Finally, Einstein offered us a broad conclusion to all his technical detail on the physical nature of space. He wrote:

“In the first place we entirely shun the vague word ‘space,’ of which, we must honestly acknowledge, we cannot form the slightest conception, and we replace it by ‘motion relative to a practically rigid body of reference.’”

In other words, space as an abstraction — or as an absolute — must be “entirely shun[ed]”.

[I can be found on Twitter here.]

When Alan Turing and Ludwig Wittgenstein Discussed the Liar Paradox

 

Alan Turing attended Ludwig Wittgenstein’s ‘Lectures on the Foundations of Mathematics’ in Cambridge in 1939. The following is one account of those lectures:

“For several terms at Cambridge in 1939, Ludwig Wittgenstein lectured on the philosophical foundations of mathematics. A lecture class taught by Wittgenstein, however, hardly resembled a lecture. He sat on a chair in the middle of the room, with some of the class sitting in chairs, some on the floor. He never used notes. He paused frequently, sometimes for several minutes, while he puzzled out a problem. He often asked his listeners questions and reacted to their replies. Many meetings were largely conversation.”

In relevance to this essay, Alan Turing (1912–1954) strongly disagreed with Ludwig Wittgenstein’s argument that mathematicians and philosophers should happily allow contradictions to exist within mathematical systems.

In basic terms, Wittgenstein stressed two things:

1) The strong distinction which must be made between accepting contradictions within mathematics and accepting contradictions outside mathematics.

2) The supposed applications and consequences of these mathematical contradictions and paradoxes outside mathematics.

As for 1) above, Wittgenstein said (as quoted by Andrew Hodges):

“Why are people afraid of contradictions? It is easy to understand why they should be afraid of contradictions in orders, descriptions, etc. outside mathematics. The question is: Why should they be afraid of contradictions inside mathematics?”

Wittgenstein can be read as not actually questioning the logical validity or status of these paradoxes and metatheorems. He was making a purely philosophical point about their supposed — and numerous — applications and consequences outside of mathematics. (These consequences — if not always applications — usually include stuff about consciousness, God, human intuition, the universe, human uniqueness, religion, arguments against artificial intelligence, meaning, purpose, etc.)

Thus Wittgenstein’s position on mathematical contradictions and paradoxes was largely down to his (as it has often been called) mathematical anthropocentrism. That is, to his belief that mathematics is a human invention. More concretely, in his “middle period” Wittgenstein stated that “[w]e make mathematics”; and some time later he said that we “invent” mathematics.

It can be seen, then, that Wittgenstein was clearly an anti-Platonist. Thus it’s not a surprise that he also said that

“the mathematician is not a discoverer: he is an inventor”.

Indeed the later Wittgenstein even went so far as to say that

“[i]t helps if one says: the proof of the Fermat proposition is not to be discovered, but to be invented”.

One other very concrete way in which Wittgenstein expressed his anti-Platonism was when he made the point that it’s wrong to assume that because

“a straight line can be drawn between any two points [that] the line already exists even if no one has drawn it”.

Wittgenstein consequently made the ironic comparison (which many may find ridiculous) that “chess only had to be discovered, it was always there!”.

In terms of contradictions and paradoxes again.

All the above means that if mathematics is a human invention, then any contradictions and paradoxes there are (within mathematics) must be down to… us. And if they’re down to us, then they aren’t telling us anything about the physical world (which includes Turing’s bridge — see later) or even about a platonic world of numbers — because such as thing doesn’t even exist.

Yet many of Wittgenstein’s remarks on paradoxes, Gödel's theorems, mathematical contradictions, etc. have been seen — by various commentators — as being almost (to use my own word) philistine in nature. (Much has been written on Wittgenstein’s remarks on Gödel's theorems — see here.)

The Liar Language Game

Wittgenstein tackled the most famous of all paradoxes — the Liar Paradox. In a discussion with Turing, he said:

“Think of the case of the Liar: It is very queer in a way that this should have puzzled anyone — much more extraordinary than you might think… Because the thing works like this: if a man says ‘I am lying’ we say that it follows that he is not lying, from which it follows that he is lying and so on. Well, so what? You can go on like that until you are black in the face. Why not? It doesn’t matter. …it is just a useless language-game, and why should anyone be excited?”

At first glance it seems that Wittgenstein was perfectly correct to use the philosophical term (his own) “language-game” to refer to the Liar Paradox — as well as to many of the other paradoxes thrown up in what’s often called the foundations of mathematics. (More correctly, these paradoxes were seen to arise within various language games.) After all, the Liar paradox is internal to a language (game) which allows such a kind of self-reference. Indeed in which other language (game) would you ever find the statement, “This sentence is false”? (Even it’s supposed everyday translation - “I am a liar” — seems somewhat contrived.) These sentences simply don’t belong to everyday languages at all. Thus they must belong to a specific technical language game. (As do, for example, Gödel sentences.)

(Of course everyday language does allow other kinds of self-reference which don’t generate — obvious? — contradictions or paradoxes; such as merely referring to oneself when one says “I am happy”.)

So Wittgenstein’s position can be summed up by saying that the Liar language game doesn’t so much as display (or spot) a contradiction or paradox — it creates one.

Wittgenstein was basically stressing the artificiality of the Liar paradox. Now that artificiality doesn’t automatically mean that it has nothing to offer us. In that case, then, the word “artificiality” needn’t be negative in tone. It may simply a reference to something which is… artificial. As it is, though, Wittgenstein did mean it in an entirely negative way. After all, he said that the Liar paradox “is just a useless language-game”.

Alan Turing, on the other hand, seemed to be interested in the Liar paradox for purely intellectual reasons. (Although he will later refer to the construction of bridges.) He replied:

“What puzzles one is that one usually uses a contradiction as a criterion for having done something wrong. But in this case one cannot find anything done wrong.”

In basic terms, Turing was arguing that, unlike many other cases of contradiction, the Lair paradox doesn’t simply uncover a contradiction: it makes it the case that both x and not-x must be accepted. That is, when a (Cretan) liar utters “I am lying”, and it leads to it being interpreted as making the speaker both a liar and not a liar (i.e., at one and the same time), then “in this case one cannot find anything done wrong”.

One can almost guess Wittgenstein’s reply to this. He said:

“Yes — and more: nothing has been done wrong [].”

Wittgenstein’s argument (at least as it can be seen) was that the Liar paradox does indeed lead to this bizarre conclusion because — in a strong sense - it was designed to do so. That is, it is part of a language-game which was specifically created to bring about a paradox. And because it’s a self-enclosed and artificial language-game, then Wittgenstein also asked “where will the harm come” from allowing such a contradiction or paradox?

Alan Turing’s Bridge

It was said a moment ago that Alan Turing appeared to be interested in the Lair paradox for purely formal reasons. However, he did then state the following:

“The real harm will not come in unless there is an application, in which a bridge may fall down or something of that sort [] You cannot be confident about applying your calculus until you know that there are no hidden contradictions in it.”

On the surface at least, it does seem somewhat bizarre that Turing should have even suspected that the Liar paradox could lead to a bridge falling down. That is, Turing believed — if somewhat tangentially — that a bridge may fall down if some of the mathematics used in its design somehow instantiated a paradox (or a contradiction) of the kind exemplified by the Liar paradox.

Yet it’s hard to imagine the precise route from the Lair paradox to practical (or concrete) applications of mathematics of any kind — let alone to the building of a bridge and then that bridge falling down.

Indeed many (pure) mathematicians have often noted the complete irrelevance of much of this paradoxical and foundational stuff to what they do. Thus if it’s irrelevant to many mathematicians, then surely it would be even more irrelevant to the designers who use mathematics in the design of their bridges.

This metamathematics/the applications of mathematics opposition is summed up by the mathematician and physicist Alan Sokal in two parts. Firstly, Sokal stresses the difference between “metatheorems” and “conventional mathematical theorems” in the following way:

“[] Metatheorems in mathematical logic, such as Gödel's theorem or independence theorems in set theory, have a logical status that is slightly different from that of conventional mathematical theorems.”

And it’s precisely because of this substantive difference that Sokal continues in this way:

“It should, however be emphasized that these rarefied branches of mathematics have very little impact on the bulk of mathematical research and almost no impact on the natural sciences.”

So if such metatheorems have (to be rhetorical for a moment) almost zero “impact on the natural sciences”, then surely they have less than zero impact on the design of bridges.

Again, it’s hard to see how there could be any (as it were) concrete manifestation of the Liar paradox. That said, perhaps Turing’s argument is that there couldn’t be such a concrete manifestation. And that’s precisely because if there were such a manifestation — then some bridges would fall down!

So what about Wittgenstein's response to this line of reasoning?

Wittgenstein responded to Turing by saying that “[b]ut nothing has ever gone wrong that way yet”. That is, no bridge has ever fallen down due to a paradox or contradiction in mathematics.

The Principle of Explosion

As already hinted at, all the above can be boiled down to Alan Turing predicting (or simply conceiving of) concrete and design-related manifestations of what is called (by logicians) the principle of explosion.

Yet it was Wittgenstein who noted what Turing was actually getting at. He said:

“Suppose I convince [someone] of the paradox of the Liar, and he says, ‘I lie, therefore I do not lie, therefore I lie and I do not lie, therefore we have a contradiction, therefore 2x2 = 369.’…”

In other words, “from a contradiction, anything follows”. Or to put that as Wittgenstein himself put it:

If we allow the sentence

“I lie, therefore I do not lie, therefore I lie and I do not lie…”

then we must also allow this equation:

2 x 2 = 369

But in the case of 2 x 2 = 369, Wittgenstein argued that “we should not call this ‘multiplication’ at all”. And surely he was right. Yet this conclusion is seen to be a logical consequence of accepting the legitimacy of the Liar paradox.

Finally, it can also be added, in a Wittgensteinian manner, that we are free to invent a language (game) in which 2 x 2 (or, perhaps more accurately, “2 x 2”) does indeed equal 369 (or “369”)!

[I can be found on Twitter here.]

Daniel Dennett on the Difference Between Imagining & Conceiving a Zombie

 

i) Introduction

ii) Descartes on Imagining and Conceiving

iii) Conceiving and Intuition

iv) What is it to Conceive of a Philosophical Zombie?

v) Philip Goff on Anil Seth Confusing Imagining and Conceiving

The philosophical notion of conceivability is at the very heart of the work of philosophers like David Chalmers and Philip Goff. Without it, their arguments against physicalism — and in support of the possibility of philosophical zombies - wouldn’t even get off the ground in the first place. In fact these philosophers wouldn’t even be widely known today if it weren’t for the importance they’ve placed on (philosophical) conceivability.

For example, in David Chalmers’ brilliant book, The Consciousness Mind (1996), there are comments and arguments about logical possibility and conceivability on almost every page. And this was the book which jumpstarted Chalmers’ career.

I even suspect that Chalmers and Goff would agree with this account of the importance they place on conceivability. That said, they probably wouldn’t word it in precisely the same in which way I have.

So it’s ironic that despite this importance given to distinguishing imagining any given x from conceiving of that same x, the philosopher Daniel Dennett (1942-) still asks the following question:

“Can you conceive of one? Can you imagine one? What is the difference?”

I doubt that Dennett is arguing that there’s no difference at all between imagining and conceiving any given x. Perhaps he’s simply claiming that it’s a difference that doesn’t really make (much of) a difference (i.e., to these specific issues).

I would also add that — in these cases at least — it’s not just that there’s no difference between conceiving and imagining any given x (or that the difference isn’t important), it’s that nothing may be conceived of in the first place — at least in the case of philosophical zombies! (See my Can You Conceive of a Philosophical Zombie… or a Million-Sided Object? | by Paul Austin Murphy | Curious | Medium.)

In addition and to use Dennett’s words, “we are entitled to ask them how they would know” that they’ve conceived of any given x.

Dennett’s question above is about those people who “say they can conceive of (philosophical) zombies”. So we can ask how they know that they’re conceiving a zombie rather than (merely) imagining one. That is, “[w]hat is the difference” between conceiving of a philosophical zombie and imagining a philosophical zombie?

It may be worse than that: perhaps neither conceiving nor imagining will help us distinguish a human being who actually instantiates experience (or consciousness) from a philosophical zombie. So whichever option we choose, there may still be a more particular problem when it comes to philosophical zombies.

(Note: If it’s accepted that imagining and conceiving are so different, then surely we have no right to say that the very same x has been both conceived of and imagined. In other words, what does the conceiving of any given x and the imagining of that same x share?)

Nearly all of this dates back to Descartes.

Descartes on Imagining and Conceiving

The 17th-century French philosopher Descartes (1596–1640) emphasised — in strong terms — this distinction between imagining and conceiving. The following is Daniel Dennett putting Descartes’ position:

“Just imagining something is not enough — and, in fact, Descartes tells us, it is not conceiving at all. According to Descartes, imagining uses your (ultimately mechanistic) body, with all its limitations (nearsightedness, limited resolution, angle, and depth); conceiving uses just your mind, which is a much more powerful organ of discernment, unfettered by the requirements of mechanism.”

So here we have a concern with distinguishing our imagining any given x and our conceiving of that same x. Just as relevantly, that distinction is tied very strongly to a dualist (or, more widely, a non-physicalist) explanation of why the two are so dissimilar. And it just so happens that the contemporary philosophers David Chalmers (1966-) and Philip Goff also incorporate dualism (or at least anti-physicalism) into their overall philosophies (more of which later).

In the case of Descartes, it’s the non-physical mind (which is a “much more powerful organ of discernment”) that allows us to conceive of x; whereas the “ultimately mechanistic body” (“with all its limitations”) allows us only to imagine that same x.

Thus the non-physical mind is necessary — at least according to Descartes — for conceiving of anything. In parallel, if we focus on (mere) imagination, then we’re only focussing on the body. And the body (with all its limitations) is not enough for the conceivings which Descartes had in mind.

So what did Descartes have in mind?

In this instance at least, Dennett tells us that Descartes

“offers a compelling example of the difference: the chiliagon, a regular thousand-sided polygon”.

And guess what — Philip Goff also focussed on polygons when he discussed conceivability. Except, in his own case, he ups the ante and cites — as a relevant example — the case of a million-sided polygon: a megagon! (See later section.)

Dennett goes into more detail about Descartes’ position on conceiving. He writes:

“Descartes doesn’t tell you to perform such constructions; to him conception, like imagination, is a kind of direct and episodic mental act, glomming without bothering to picture, or something like that.”

Basically, the best (if somewhat oxymoronic) way of putting Descartes’ position is to say that the conceiving (or “conception”) of any given x is an act of imagination which doesn’t actually use (or include) any mental images!

This raises the question:

Once we take away all the (mental) images, then what do we have left?

Conceiving and Intuition

Dennett’s description of Descartes’ position makes it seem that the conceiving any given x is very similar to the philosophical notion of intuiting (or having “direct insight” into) such an x. This is especially the case when Dennett says that these kinds of conceiving are examples of “glomming without bothering to picture”.

Thus when Descartes conceived of a chiliagon it might have been like a (as it were) Gödelian mathematician intuiting the truth of an unprovable statement (see here). And consider too a mathematician like Roger Penrose (1932-) who plays down “picturing” and even “words” when it comes to gaining access to mathematical truths in the platonic realm. (Penrose also plays down words and pictures when he does mathematics generally.) For example, Penrose also seems to go beyond purely mathematical Platonism when he stated the following:

“[I] find words almost useless for mathematical thinking. Other kinds of thinking, perhaps such as philosophizing, seem to be much better suited to verbal expression. Perhaps this is why so many philosophers seem to be of the opinion that language is essential for intelligent or conscious thought!”

Alternatively, Descartes’ conceivings might have been like the philosopher Laurence BonJour’s own direct insights into metaphysical necessities — those necessities which are true of the physical world itself.

Basically, then, Dennett’s account of Cartesian conceiving (i.e., as a “kind of direct and episodic mental act”) seems like a perfect description of an act of intuition or direct insight.

That said, Dennett’s final words are perhaps the most important (or relevant) to this discussion. (After all, much has already been written on Gödelian and other kinds of intuition.) Dennett concludes:

“You somehow just grasp (mentally) the relevant concepts (SIDE, THOUSAND, REGULAR, POLYGON), and shazam! You’ve got it. I have always been suspicious of this Cartesian basic act of conceiving.”

Now what exactly is it to “grasp (mentally)” the concepts SIDE, THOUSAND, REGULAR and POLYGON? In addition, don’t these concepts need to be (as it were) fused together in order to grasp a chiliagon? After all, whatever conceiving of the concept SIDE, THOUSAND, REGULAR or POLYGON separately consists in, these concepts still need to be stuck together in order to grasp the broader concept — CHILIAGON.

What is it to Conceive of a Philosophical Zombie?

Perhaps Descartes was on much stronger ground when he discussed the conceiving of a chiliagon than David Chalmers and Philip Goff are when they discuss the conceiving of a philosophical zombie. Or the very least that can be said is that a chiliagon is in a different logical space to a philosophical zombie.

But, again, what is it to conceive of a philosophical zombie? What is the mental or abstract content of such an act of conceiving?

Dennett picks up on this in the following way:

“When people say they can conceive of (philosophical) zombies, we are entitled to ask them how they know. Conceiving is not easy!”

More particularly, how does, for example, a dualist, anti-physicalist or anyone else know that he’s conceived of a philosophical zombie? How do we know that he has conceived of a philosophical zombie? In addition, how does he (to use Michael Dummett’s term) “manifest” his act of conceiving of a zombie to others? What if it’s a thoroughly private act? And, if it is private, then what status could it possibly have when it comes to establishing a metaphysical position or thesis?

We can get even more fundamental here: What is it to conceive of… anything? This isn’t to argue that we don’t conceive of things. It’s just a demand for some kind of account.

In any case, Dennett give some examples of things which he believes are difficult to conceive. He writes:

“Can you conceive of more than three dimensions? The curvature of space? Quantum entanglement?”

The least that can be said is that Dennett’s examples are all very different.

So perhaps we can’t imagine more than three dimensions, the curvature of space and quantum entanglement; though we can conceive of them.

Dennett continues (as already quoted):

“Just imagining something is not enough — and, in fact, Descartes tells us, it is not conceiving at all.”

The thing is that the existence of more than three dimensions, the curvature of space and quantum entanglement must have been conceived of — many times — because they’re accepted notions in physics. Indeed they’re even accepted aspects of the physical world (or at least two of them are)! That is, no one has ever seen or observed these things. And, depending on definitions, not one has ever imagined these things either. So all we have left is to conceive of more than three dimensions, the curvature of space and quantum entanglement.

Thus Descartes, Goff and Chalmers may be onto something here!

Yet even here conceiving of these things may be in a different logical space to conceiving of a philosophical zombie. After all, there are a lot of equations, natural laws, theories, indirect/direct observations, experiments, etc. to account for extra dimensions, the curvature of space and quantum entanglement. Are there a lot of equations, natural laws, theories, experiments, indirect/direct observations, experiments, etc. to account for philosophical zombies?

Of course not.

And that’s primarily because Goff and Chalmers themselves accept that philosophical zombies are only a logical possibility. Thus there are no equations, direct/indirect observations, natural laws, experiments or physical theories which account for the existence of philosophical zombies.

Philip Goff on Anil Seth’s Confusing Imagining and Conceiving

Anil Seth

As stated, both David Chalmers and Philip Goff make much of the distinction that we must make between (merely) imagining x and conceiving of that same x. To them, this difference is extremely important.

For example, the following is Philip Goff writing about those academics who confuse (or conflate) the two:

“The zombie argument is generally known in the academic philosophical literature as the ‘conceivability argument.’ I think this is something of a misnomer, as it suggests that the argument has something to do with what can be imagined.”

So not only does Goff believe that the word “imagine” is misleading: he also believes the same about the word “conceive. Of course that’s primarily because Goff believes that people conflate (or confuse) imaginability with conceivability.

Philip Goff

As it is, Goff doesn’t care that much about merely imagining any given x: his position is about the conceiving of that x.

(Again: if it’s accepted that imagining and conceiving are so different, then surely we have no right to say that the very same x has been both conceived of and imagined. In other words, what does the conceiving of any given x and the imagining of that same x share?)

So Goff spots such a confusion (or conflation) in the arguments of the cognitive neuroscientist Anil Seth (1972-).

Goff firstly quotes Anil Seth’s own words in the following way:

“‘Conceivability arguments are generally weak since they often rest on failures of imagination or knowledge, rather than on insights into necessity. For example: the more I know about aerodynamics, the less I can imagine a 787 Dreamliner flying backwards. It cannot be done and such a thing is only ‘conceivable’ through ignorance about how wings work.’”

And the following is Goff’s response to that passage:

“The zombies argument is concerned with logical possibility, whereas Seth’s example deals with natural possibility. It is inconsistent with the laws of nature for a 787 Dreamliner to fly backward, and one appreciates this as one learns about the relevant laws of nature. But it is certainly not contradictory for a 787 Dreamliner to fly backward; if the laws of nature had been very different, such a thing might have been possible. In other words, a 787 Dreamliner flying backward is not naturally possible but it is logically possible.”

Goff derives what he calls a “logical possibility” (see here) from what he has conceived. And, in this case at least, what he conceived didn’t abide by “natural possibility”. In other words, Goff’s conceivable x outruns the natural. Indeed his conceivable x even outruns the actual (or the real).

Conclusion

Let’s forget about what is logically possible for the moment because — to Goff — it’s a product of what is conceivable. This means that the first port of call is the act of conceiving of any given x.

Yet the case against being able to conceive of a philosophical zombie has little (though not nothing) to do with with a belief that mental images (or imagery of whatever kinds) are required.

Take the case of a megagon again.

No one can imagine what a megagon looks like because it looks like a circle. The lack of conceivability in this case is down to not knowing all the mathematics. (Thus non-mathematicians must rely on the testimony of mathematicians when it comes to the — abstract — existence of a megagon; and the same goes for Daniel Dennett’s earlier examples of curved space and entanglement.)

This situation may well be passed on — at least to some extent — to the case of a philosophical zombie.

Simply writing the words “A zombie is exactly like a human being in every respect — except it has not consciousness”, and then thinking about those words and noting that they don’t contain a contradiction, is not to actually conceive of a philosophical zombie at all.

Philip Goff particularly makes the philosophical notion of conceivability seem purely logical in nature. That’s why he often mentions “contradictions”. Yet it can’t all be purely logical. Using words like “zombie”, “human being”, “experience”, “metaphysics”, “behaviour”, etc. means that Goff has automatically gone way beyond (pure) logic. In other words, Goff is using terms which are extremely loaded — from a philosophical point of view. And it can therefore be argued that a purely (as it were) logical conceiving (or reading) of a philosophical zombie no longer does the trick.

Yet Goff is still attempting to make it seem that what he’s arguing is purely logical. If it were purely logical (say, only a matter of Ps, Qs and logical operations), then a pure conceiving of a (logical) x would be fine. But we’re supposed to be conceiving of philosophical zombies!

So has Philip Goff actually conceived of a philosophical zombie (or a megagon) in the first place?

*) See my Can You Conceive of a Philosophical Zombie… or a Million-Sided Object? | by Paul Austin Murphy | Curious | Medium

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