Can You Conceive of a Philosophical Zombie… or a Million-Sided Object?

i) Introduction

ii) Conceivability

iii) Philosophical Zombies

iv) Four Conceivings: (A Round Square, Descartes’ Chiliagon, Goff’s Megagon and Water Being XYZ)

v) Saul Kripke on Misconceivings

vi) Conclusion: Philip Goff on Misconceivings

Conceivability has played a vital part in much contemporary (mainly analytic) philosophy. This is the case specifically in metaphysics. It’s also very important to the work of the Australian philosopher David Chalmers and his championship of what came to be called “philosophical zombies”.

Much of this interest in conceivability begin with the work of Saul Kripke in the 1960s and 1970s. Yet Kripke himself acknowledged the possibility of people misconceiving things which they falsely believe they’ve actually conceived. And David Chalmers too acknowledges this possibility of misconceiving (metaphysical) possibilities.

But the first thing which needs to be done is to distinguish conceiving from imagining. This is an important distinction which many philosophers — since Descartes — have often stressed. Here’s one account of that distinction:

“[Descartes] first distinguishes between imagination and pure understanding. In the case of a triangle, he can perceive that a triangle is three-sided and derive all sorts of other properties using the understanding alone. He can also perceive these properties with the imagination, by picturing the triangle in his mind’s eye. However, the weaknesses of the imagination become clear when he considers a thousand-sided figure. It is very difficult to picture it in his mind’s eye, and more difficult still to differentiate that mental image from the mental image of a 999-sided figure. The pure understanding, however, dealing only in mathematical relations, can perceive all the properties of a thousand-sided figure just as easily as it can a triangle.”

As can be seen, the words above are all indebted to Descartes’ distinctions between imagining and conceiving (or “pure understanding”); as are the distinctions of David Chalmers and Philip Goff which follow.

Conceivability

To put it basically, some philosophers argue that (philosophical) conceivings don’t depend on the formation of any mental images whatsoever. Therefore such conceivings can be seen as being a more sophisticated form of …. well, for want of a better word, imagining. That is, conceiving x is (at least in part) imagining x without employing any mental imagery. (If that isn’t in itself an oxymoron.)

The general philosophical argument is that if we can conceive of any given x, then that x is possible. (Distinctions between logical possibility and metaphysical possibility will be made later.)That is, the fact that we can conceive of x, means that x is possible.

But why on earth should that be the case?

(Note: Most of the following quotes from Philip Goff are taken from his paper ‘The Phenomenal Bonding Solution to the Combination Problem’.)

In more technical terms, the English philosopher Philip Goff (more of whom later) expresses the relation between conceivability and possibility in this way:

“If P is conceivably true, then P is possibly true.”

This is also expressed in possible-worlds jargon as follows:

“If P is conceivably true (upon ideal reflection), then there is a possible world W, such that P is true at W considered as actual.”

Or, less technically, Goff also that

“Chalmers holds that every conceivably true proposition corresponds in this way to some genuine possibility”.

Goff adds that

“conceivability entails possibility when you completely understand what you’re conceiving of”.

And then Goff goes much further than all the above.

Not only is his argument that the conceiving of x is a reason for believing that x is metaphysically possible, Goff also argues that it may be the case (or is the case) that

“metaphysical possibility is just a special kind of conceivability”.

Note the use of the (possible) “‘is’ of identity” in the sentence above. We’re told that metaphysical possibility literally is (“a special kind of”) conceivability. Thus it’s not just that our conceiving of x may — or does — give us a reason to believe that x is possible. It’s also the case that this very conceiving of x seems to (as it were) bring about the metaphysical possibility of x.

Despite all that, Goff seems to assume that there’s a determinate and precise meaning of the words “conceivably” and “conceivably true”. Yet conceivability-to-possibility arguments may not get off the ground (at least in some cases) in the first place. And that’s because nothing at all is really conceived. Alternatively, that which is conceived of is literally unbelievable (as Goff himself acknowledges later).

So it can be argued that the least a philosopher can argue is that conceivability is a (rough) guide to possibility. This lesser claims would an alternative to outrightly stating that because subject S has conceived of any given x, then that means that x is possible. Of course we’ll now need to know what the words “guide to possibility” mean and how, exactly, conceiving of x is a guide to its possibility.

So let’s be more concrete about conceivability by bringing in philosophical zombies.

Philosophical Zombies

What is a philosophical zombie?

This is one account:

“A philosophical zombie or p-zombie is a hypothetical being that is physically identical to and indistinguishable from a normal person but does not have conscious experience, qualia, or sentience… Proponents of philosophical zombie arguments, such as the philosopher David Chalmers argue that since a philosophical zombie is by definition physically identical to a conscious person, even its logical possibility would refute physicalism, because it would establish the existence of conscious experience as a further fact.”

How would a philosopher (in a strict sense) know that he’d conceived of a philosophical zombie? What if you or I can’t conceive of a phil-zombie even if other people can do so? And even if you and I did (seemingly) conceive of a p-zombie, then how could we then know that we’d done so? Indeed wouldn’t the best philosophical policy be to accept (or believe) that we hadn’t actually conceived of a p-zombie?

Isn’t the very notion of conceiving of any given x far too psychologistic (or mentalistic) an idea to be sustaining metaphysical theses? Moreover, if other minds are a philosophical “problem”, then other minds conceiving of p-zombies (or anything else) is a problem within an already-existing problem.

More concretely, if we actually came across an entity which acted and looked — in every single detail — like an ordinary human being (which is the whole point of the p-zombies thought experiment), then how could we extract from all this that it may, in fact, be a p-zombie? If we ever did come across a p-zombie which acted and looked just like us, then the obvious conclusion would be that it had consciousness (or experiences) just like us. In other words, we’d assume that this “zombie” felt pain, sometimes gets angry, could smell garlic, etc.

Moreover, when Mary supposedly conceives of a p-zombie, how does it differ from her conceiving of her mother, father or best friend? After all, when she conceives of the latter, she can’t literally conceive of his/her consciousness or of them actually being conscious. Instead, all she’s got to go on is their physical and verbal behaviour. And that’s also true of p-zombies.

Perhaps that’s Chalmers’ point!

If it is Chalmers’ point, then let’s spell it out again.

Conceiving of a p-zombie is no different from conceiving of your mother, father or best friend.

So conceiving of a p-zombies is no different from conceiving of your mother, father or best friend and perhaps that’s partly why Chalmers believes conceiving of a p-zombie is possible. And if conceiving of p-zombie is possible, then its actual existence is (metaphysically) possible.

But does all that actually work?

When we conceive of a p-zombie (just as when we conceive of our mother, father or best friend), we don’t conceive of an entity (or being) actually lacking consciousness because such a thing can’t be conceived. We’re supposed to be capable of conceiving of a being (or entity) behaving like our mother, father or best friend — though not instantiating consciousness. However, how is that different to conceiving of one’s mother, father or best friend? And it’s no different because we can neither conceive of the consciousness of a fellow human being nor of a p-zombie. In other words, consciousness is “known” because it’s behaviourally or physically expressed. Unexpressed consciousness, on the other hand, is not conceivable. So it’s just as hard to conceive of another being’s consciousness as it is to conceive of its lack of consciousness.

Of course Chalmers — for one — never really discusses coming across real (or actual) p-zombies because the whole point of his thesis is the importance of conceiving of them. He’s also keen to show us how that very conceiving establishes certain other(!) philosophical positions. In this case, that other philosophical position is that consciousness (or experience) is over and above everything physical and functional. In other words, even though this conceived of (or real) p-zombie behaves and looks just like a normal human being, it still may not (or does not) instantiate consciousness at all. This means that, for all intents and purposes, a p-zombie is a (as it were) robot or machine in human clothing.

Despite all the above, can the contrary ever be categorically stated?

Can it be stated that no one could ever conceive of a p-zombie (or a “p-zombie world”)? In other words, if Chalmers, etc. can’t categorically claim that they have conceived of a p-zombie, then perhaps we can’t categorically claim that they haven’t done so. How could you, I or anyone else know what Chalmers has (or hasn’t) conceived of? Consequently, this may mean that all the sceptical questions aimed at such conceivings can now be aimed at the sceptics about such conceivings.

Yet that last conclusion surely demonstrates the central point of this issue.

That point being that since we’re talking about mentalistic conceivings, whether subject S has (or hasn’t) truly conceived of a p-zombie can never be established. And if it can never be established, then how can such conceivings provide the basis of philosophical theses?

This is a good place to look at some other (philosophical) conceivings.

Four Conceivings:

A Round Square

Here’s an extreme example of the conceivability-to-possibility argument:

i) If we can conceive of a round square,

ii) then it is possible that there is a round square.

This is a rejoinder to that argument:

i) We cannot conceive of a round square.

ii) Therefore round squares are impossible.

But what about this? -

i) If we can conceive of p-zombies,

ii) then p-zombies are possible.

Here’s the counter-argument:

i) We can’t conceive of p-zombies.

ii) Therefore p-zombies are impossible.

Of course the examples of a round square and a p-zombie are very different (as are a megagon and p-zombie). However, this round square example was chosen simply to get the point across.

In other cases, the move form conceivings to metaphysical possibilities may well be legitimate. Alternatively, every move from conceivability to metaphysical possibility may be somewhat suspect!

Descartes’ Chiliagon

This is an artistic representation of a chiliagon. A chiliagon doesn’t actually look this this — because it doesn’t really look like anything (i.e., other than a circle.)

The definitional quote in the introduction above mentions a “999-sided figure”; whereas René Descartes himself referred to a chiliagon - which is a 1000-sided polygon. (I presume the odd and arbitrary number of 999 sides — rather than the 1000 sides of a chiliagon — was chosen to emphasise our imaginative limitations.)

Even if a chiliagon were massive in size, it would still be visually indistinguishable from a circle. It can also be argued that a chiliagon can’t be conceived of either — even if we do have a concept of it. Though that, of course, will depend on what’s meant by the words “conceived of”. It can be argued that mathematicians (or geometricians) can conceive of a chiliagon (to lesser or greater degrees); though a layperson can’t.

Moreover, a chiliagon is a mathematical/geometrical abstract object; not a concrete object. In other words, it couldn’t be found or even made. Indeed both the chiliagon and megagon are not a constructible polygons. Nonetheless, that doesn’t stop them from being well-defined concepts.

Philip Goff’s Million-Sided Object

This is an artistic representation of a megagon. A megagon (like a chiliagon) doesn’t actually look this this — because it doesn’t really look like anything (i.e., other than a circle.)

Philip Goff himself cites the example of a “million-sided object” to stress the distinction between conceiving and imagining any given x.

The megagon (a million-sided polygon) is obviously similar to Descartes’ example of the chiliagon in that it’s deemed to be a “well-defined concept” which, nevertheless, cannot be visualised.

In one sense it can be said that we can indeed conceive of a megagon. Or, more helpfully, if I were to ask someone this question:

What do you conceive of when you say that you’ve “conceived of a million-sided object”?

That person may answer:

I conceived of an object which has a million sides.

But what does that mean? What is he conceiving of? Is he simply assuming the following? -

i) A million-sided object has a million sides.

ii) Therefore I’ve conceived of a million-sided object.

Is there any more to it than that? If he simply knows that if an object has a million sides, then he’s conceived of an object which has a million sides?

In any case, no one can picture or imagine a million-sided object. So that’s ruled out for a start. What’s left? Again, the words “conceiving a million-sided object” may seem vacuous. Yet despite all that Goff says that “the concept million-sided object is transparent”. That is,

“it is a priori (for someone possessing the concept, and in virtue of possessing the concept) what it is for something to have a million sides”.

Moreover,

“when one conceives of a million-sided object one completely understands, or is in principle able to reason one’s way to a complete understanding of, the situation being conceived of”.

Goff goes further when he says that

“it is a priori for the conceiver what it is for the state of affairs they are conceiving of [i.e., a million-sided object] to obtain”.

Thus we reach the important conclusion which Goff has been leading up to all along. Namely,

“that we can move from the conceivability (upon ideal reflection) of the states of affairs so conceived, to its genuine possibility”.

Goff’s quote above is simply a rerun of what’s already been said. That is:

i) What is it to conceive of something which has a million sides?

ii) It is to conceive of a million-sided object.

Nonetheless, perhaps my position is too psychologistic in nature — i.e., too dependent on our contingent mental states and their content (as Descartes warned us against). Perhaps Goff’s position is purely logical. Alternatively, perhaps Goff’s position is purely mathematical/geometrical (therefore abstract) in nature. That would mean that it may be an entirely logical and/or mathematical point to say that “the concept million-sided object is transparent”.

Then again, what does that claim amount to? Indeed how different is conceiving of a million-sided object to conceiving of a round square? That said, it’s clearly the case that a round square isn’t in the same logical space as a million-sided object. A round square is a logically impossible object. (Or is it conceptually impossible?) A megagon, on the other hand, is not a logically impossible object — even though it isn’t a constructible polygon.

Moreover, even if the words “having a well-defined concept” and “conceiving of” are taken as (virtual) synonyms, it’s still the case that both the layperson and the mathematician would need to conceive of (or have a well-defined concept of) a megagon.

So can Goff — or any other philosopher who isn’t a mathematician — really conceive of a megagon without also (to use Goff’s own words) “completely understanding” all the detail in the following copy-and-paste? Here goes:

The question now is:

Are p-zombies in the least bit analogous to a million-sided polygon (i.e., a megagon) when it comes to conceivability leading to possibility?

Water is XYZ

David Chalmers also offers us an example of conceiving of water being XYZ rather than H₂O.

Again, what does it mean to conceive that water is XYZ?

Surely it’s no use Chalmers going into to further detail if this isn’t established in the first place.

Is water being XYZ conceivable simply because we’re simply imagining (to use (Chalmers’ own words) “watery stuff” (i.e., Chalmers’ “primary intension”) which we also conceive to be XYZ? But are we conceiving of water actually being XYZ (i.e., rather than simply conceiving of water stuff)? Isn’t that something completely different? So here goes:

1) The first act of conceiving has to take on board what XYZ actually is. (Is XYZ meant to be some kind of fictional — though possible — molecule?)
2) And then one needs to conceive of XYZ actually being water.

Colorless Green Ideas Sleep Furiously

Now take Chomsky’s famous surreal sentence:

“Colorless green ideas sleep furiously.”

All the words and their concepts in the sentence above are (to use Goff’s word) “transparent” when taken individually. We can also say that the sentence itself is grammatically and logically correct. We can understand the words within that statement. However, can we also conceive of that sentence being true? Or, more accurately, can we conceive of a (to use Goff’s words) “state of affairs” in which colorless green ideas sleep furiously?

Here it seems that grammatical (or even logical) correctness runs free of conceivability. In other words, perhaps we don’t — and can’t — actually conceive of colorless green ideas sleeping furiously.

But is the same conclusion true of conceiving of a p-zombie?

Saul Kripke on Misconceivings

The American philosopher and logician Saul Kripke cited various examples of people believing that they’ve conceived of something — which they deemed to be possible - and their not actually doing so.

Kripke offered us the example of conceiving that water is not H₂O. He also cited the case of a person seemingly conceiving of a true mathematical theorem to be false or false theorem to be true. (In the former case, Kripke gave the example of Goldbach’s conjecture.)

Kripke — like Chalmers and Goff — claimed that he was working with his own “Cartesian intuitions” when he — specifically — tackled the mind-body problem. And many of those intuitions were also about what is and what isn’t metaphysically possible.

In the first instance, Kripke cited an act of imagination (i.e., he didn’t use the word “conceiving”) which may mislead us. He wrote:

“[W]e thought erroneously that we could imagine a situation in which heat was not the motion of molecules. Because although we can say that we pick out heat contingently by the contingent property that it affects us in such and such way.”

In this case “we” conceived of the effects of the motion of molecules on bodies and the environment. However, we didn’t — or couldn’t — conceive of heat actually being something other than the motion of molecules.

So let’s say that heat is XYZ and that it has nothing to do with molecular motion. That raises the following question:

What is it to conceive that heat is XYZ rather than molecular motion?

Kripke was claiming that those who imagine heat being something that’s not molecular motion aren’t really imagining heat at all. They just believe that they are imagining heat because they’re actually imagining the “contingent properties” of heat — i.e., its effects on persons and the environment.

To repeat: we can’t conceive of such a thing as heat being XYZ. In other words:

1) The argument is not only that there is a radical separation between our conceiving of XYZ being heat (or conceiving p-zombies) and its actual possibility.

2) The argument is also that we don’t actually conceive of such a thing in the first place.

Yet Kripke also argued that our imagination can tell us something both true and important about the world. In Kripke’s own words:

“[J]ust as it seems that the brain state could have existed without any pain, so it seems that the pain could have existed without the corresponding brain state.”

Kripke stressed our ability to imagine a pain state without its correlated brain state (formerly characterised as the “firing of C-fibres”). Thus Kripke (to paraphrase) concluded:

If we can imagine a mental state without its correlated brain state, then such a states of affairs is possible.

Alternatively, Kripke was also arguing that there’s no necessary identity between mental states and brain states.

Conclusion: Philip Goff on Misconceivings

Philip Goff also expresses the possibility that conceivability may not always give us metaphysical (i., not logical) possibility. That is, even if we do allow various moves from conceivability to metaphysical possibility, sometimes what we argue is metaphysically possible still remains unbelievable. Or as Goff himself puts it:

“When metaphysical possibility is so radically divorced from conceptual coherence [] I start to lose my grip on what metaphysical possibility is supposed to be.”

Goff is arguing that — in some cases at least — metaphysical possibility can move too far beyond “conceptual coherence”. That must also mean that — at least some — moves from conceivability to metaphysical possibility are illegitimate. That is, a specific conceiving may not warrant the metaphysical possibility which is supposedly derived from it. To stress that point, Goff also says that

“a radical separation between what is conceivable and what is possible has the potential to make our knowledge of possibility problematic”.

Yet, despite all the above, both Goff and Chalmers seem to provide what they take to be a tight link between conceivability and metaphysical possibility. And, if that’s the case, then how can there ever be a (to use Goff’s words) “radical separation” between the two?

That said, the central position taken in this piece is that if that link between conceivability and metaphysical possibility is — at least in some cases — broken, then that will be (at least partly) due to the fact that such conceivings aren’t really genuine conceivings at all. More clearly, some moves from conceiving to (metaphysical) possibility are completely bogus precisely because the given x wasn’t conceived of in the first place.

 

R.L. Wilder’s Constructivist Account of Early 20th Century Mathematics

 

This is the second part of my piece on Raymond Louis Wilder (1896–1982) and his philosophical, historical and anthropological account of mathematics. I suggest that the reader refer back to the introduction to my ‘Raymond L. Wilder’s Anthropology of Mathematics: Platonism and Applied Mathematics’ for Wilder’s biographical details.

Introduction

R.L. Wilder called for mathematics to be analysed by the social sciences. He suggested that we should

“study mathematics as a human artefact, as a natural phenomenon subject to empirical observation and scientific analysis, and, in particular, as a cultural phenomenon understandable in anthropological terms”.

Moreover, Wilder wrote the following words:

“The major difference between mathematics and the other sciences, natural and social, is that whereas the latter are directly restricted in their purview by environmental phenomena of a physical or social nature, mathematics is subject only indirectly to such limitations.”

What follows is primarily a commentary on Wilder’s well-known book Evolution of Mathematical Concepts: An Elementary Study; which was written in 1968.

R.L. Wilder’s General Account of Mathematics

R.L. Wilder offered us a constructivist and anthropological view of mathematics which is radically at odds with the general Platonic (or at least quasi-Platonic) position.

Wilder himself wrote:

“Mathematics derives its concepts initially from the existing world of reality and uses them as a way of dealing with this reality.”

The traditional view is that it’s indeed the case that mathematics can be used “as a way of dealing with” reality”. However, it’s not usually also said that mathematics actually “derives its concepts [] from the existing world of reality”. Indeed it’s quite hard to see how mathematics can derive its concepts from reality without mathematicians already having at least some mathematical concepts to begin with. That said, if mathematical concepts are derived from reality (if only initially), then it’s no surprise that they can then be applied to — or deal with — that reality.

Now here’s the English cosmologist, theoretical physicist and mathematician John D. Barrow expressing the same position as it was advanced by the 19th century Dutch mathematician Diederik Korteweg:

“Korteweg’s own philosophy of mathematics was a straightforward one. He believed that we had discovered mathematics from the physical world and so its applicability there was just following the stream back to its source.”

Following on from Barrow’s words, there’s an interesting qualification — if it is a qualification — to what Wilder wrote above. Wilder continued by saying that this reality

“embraces not just the physical environment, but the cultural — which includes the conceptual — environment”.

In fact Wilder claimed that mathematical concepts “are just as real as guns or butter”. The point being made here is that human (even if deemed to be abstract) concepts don’t run free of the “physical environment”. And this is something that mathematical — or any other kinds of — Platonists won’t accept.

Wilder then explicitly put the anti-Platonist position. He argued that the concepts of mathematics

“were no longer embodiments of an independently existing realms of ideas, having an existence before and after the fact of their discovery, but only of a world of concepts continually under construction and having no existence until conceived in the minds of the mathematicians who created them”.

This is pure mathematical constructivism.

Wilder on Logic, Proof and Set Theory

Wilders stated that “the Greeks brought the notion of proof by logic into mathematics”. That said, many people still associate proof with mathematics.

Now take the well-known logical laws of contradiction and excluded middle.

It wasn’t the case that the law of non-contradiction and the law of excluded middle had logical proofs themselves. Instead, their “trustworthiness” in mathematics (according to Wilder) “was not questioned”. In addition, the mathematical

“conclusions reached by the use of such ‘laws’ were considered absolutely reliable if the premises were”.

Thus the law of contradiction and the excluded middle were essentially used as logical axioms in mathematics. Indeed these laws (or axioms) were deemed to be at the very basis (or foundation) of all mathematics - at least until the late 19th century.

So what about sets?

Wilder told us that it was “nineteenth-century mathematicians [who] introduced set theory into mathematics”. Wilder then provided a constructivist — and “materialist” (his own word) — account of set theory. He went on to say that set theory

“was derived from experience with the finite collections of the physical and cultural environments”.

This is no surprise if we view sets as simply being collections of their concrete members. That is, if we see sets as being definable exclusively in terms of their members. However, this position leads to problems if it truly is all about “the finite collections of the physical environment”.

In other words, what about the null set and “infinite domains”?

Wilder continued:

“That extension of the classical logic and of set theory to infinite domains might lead to difficulties was not generally anticipated until around 1900, when a number of contradictions were found.”

Because infinite sets led to contradictions, then it was no wonder that the English philosopher Bertrand Russell (at least at one time) argued that sets are nothing over and above the sum of their members. Clearly this makes less sense when we take into consideration infinite sets and the null set. What are the members of such sets? How do we count them? How can an infinity determine a (circumscribed) set at all?

It was because of these problems — i.e., at the end of the 19th century — that commentators began to speak in terms of the “crisis in mathematics”. And, in order to end that crisis, a

“new foundation for the whole of mathematics seemed necessary to meet this crisis, not just a revised formulation of the real number continuum; for all parts of mathematics depended to a greater or lesser extent on logic and set theory”.

This explains the obsessive search for foundations in mathematics in this period; as can be seen in Frege, Russell and many other philosophers, logicians and mathematicians. It also shows us the importance of logic — in the guise of set theory — to mathematics. (Incidentally, similar “crises” have also occurred in philosophy; such as the case with epistemology and its equivalent search for foundations. See epistemological foundationalism.)

One of the best known (at least to philosophers) quests for the foundations of mathematics can be found in the work of Gottlob Frege. According to Wilder, Frege

“insisted that number and all of mathematics can be grounded in logic — a doctrine sometimes called ‘the logicist thesis’”.

From what’s been said about set theory’s importance to 19th-century mathematics (indeed, all mathematics), it isn’t surprising that Frege attempted to reduce mathematics — beginning with arithmetic — to logic. Wilder also tells us that Giuseppe Peano attempted something similar when he “refined and utilised the axiomatic method to achieve a basis for mathematics”.

A decade or so later, it was “largely under the influence of the works of Frege and Peano that the work of Russell and Whitehead was fashioned”. This too was a logicist programme. More technically, in the Principia Mathematica “an attempt was made to derive mathematics from self-evident universal (‘tautological’) logical truths”.

Despite the above, it’s not the case that logical tautologies (or logical truths) are required to be “self-evident” or even evident in nature. (These notions belong to what philosophers call psychologism.) What they need to be is necessarily true. As Ludwig Wittgenstein later put it, tautologies don’t even allow the possibility of their own falsehood. Thus their necessary truth is a result of their form, not of their content (see here).

I’ve just mentioned that tautologies needn’t be self-evident in nature. This lack of self-evidence contributed to the problems which irked the logicist programme. Wilder went on to say that as this foundationalist

“work proceeded to the higher realms of mathematical abstraction, it became necessary to introduce axioms that could hardly be admitted as constituting ‘self-evident logical truths’…”

Again, this raises the following question.

What did these philosophers, logicians and mathematicians mean by the term “self-evident”?

The problem here is that what’s self-evident to a mathematician may not be self-evident to a layperson. What’s more, what’s self-evident to the higher-level mathematician may not be self-evident to the lower-level mathematician. Indeed if a logical truth (or axiom) is necessarily true, then why do we need the added property self-evidence at all? And can a logical truth (or equation) simply (as it were) become self-evident after the mathematician (or logician) has worked on it for some time? That said, would this work on self-evidence constitute some kind of contradiction (or negation) of something’s being self-evident?

Kronecker, Intuitionism & the Law of Excluded Middle

According to Wilder, the German mathematician Leopold Kronecker fiercely derided Georg Cantor’s infinities. That was mainly because Kronecker believed that mathematics

“was a construction based on the natural numbers, which, in turn, were an outgrowth of man’s ‘intuition’”.

So not only do we have a reference to “construction” here: we also have a reference to “intuition”.

We can now ask why the natural numbers are so special and why they too aren’t constructed. In addition, what did Kronecker mean by the word “intuition”? (This term was initially taken from Immanuel Kant’s philosophy of mathematics — see here.)

Wilder then went into more detail as to what it was that Kronecker believed. He stated that Kronecker

“avoided all use of numbers that could not be constructed (as can, for instance, fractions like 2/3) from natural numbers”.

We now need to ask what exactly is meant by the words “constructed from natural numbers”. This — at least partly — means that numbers are constructed by use of operations such as +, x, etc. In any case, Kronecker

“asserted that numbers like [x] for example, simply do not ‘exist’, since there are apparently no ways of constructing them from natural numbers”.

We can say, then, that Kronecker was a constructivist — but only about certain numbers.

I mentioned Kronecker’s acceptance of natural numbers earlier and he accepted them because they are… well, natural.

Wilder went on to say that “[v]irtually no one agreed with” Kronecker on these issues.

Kronecker can also be seen as a kind of (proto)intuitionist. Wilder said that his “thesis was reaffirmed (in modified form) by Brouwer”. Wilder continued:

“[L]ogic that had been introduced into mathematics by the Greeks was tossed overboard, except for what could be salvaged through use of the constructive methods of intuitionism.”

What, exactly, was thrown overboard? Perhaps, most importantly, the

“use of the law of the excluded middle, so important in reductio ad absurdum proofs, was no longer permissible except for finite sets”.

Interestingly enough, Wilder claimed that the law of excluded middle is acceptable to the intuitionist. However, this was the case only when that law was applied to finite sets — which is, of course, a big exception! Wilder did go on to tell us why the law of excluded middle is applicable to finite sets when he said that for

“any finite set of natural numbers, it was permissible to assert that either at least one of the numbers was even or none was even”.

For intuitionism, what is important is how a mathematical statement is proven. That is, there must

“exist an elementary constructive was of demonstrating such use of the law of the excluded middle, namely by examining the numbers one by one!”.

So here we have an explicit example of intuitionist construction — “examining the numbers one by one”. That is: Is the number 1 even? No. Is the number 2 even? Yes. Okay. That means that there is at least one even number in this set of two natural numbers.

So what is it about infinite sets that renders the law of excluded middle non-applicable?

We can say that

“the same assertion about an arbitrary infinite set of natural numbers could not be made”.

Why is that? It’s because we can never know what surprises an infinite set contains or will (as it were) throw up later.

For example, we may count a billion billion numbers of an infinite set and find that, so far, it doesn’t instantiate property x. However, that doesn’t mean that it may not display x somewhere further up the line. Thus we can’t apply the law of excluded middle (i.e., either p or not-p) to this particular case.

So what is positive about intuitionism? What does it (or did it) attempt to achieve?

Wilder went on to say that the

“great advantage of intuitionistic philosophy was its freedom from contradiction — limitation to constructive methods guaranteed this”.

The idea here is that if one is constructing everything (as it were) by hand , then one can’t make mistakes. In addition, there’s no room for speculation or conjecture in intuitionism — and that’s why it is (supposedly) guaranteed to be free from contradictions. In other words, all such contradictions will be nipped in the bud.

Of course intuitionism does have its defects. Wilder did say that its

“fatal defect was that it could not derive, using only constructivist methods, a major portion of the concepts that were regarded as being among the greatest mathematical achievements of the modern era”.

Perhaps many of these great mathematical concepts were the result of speculation or mathematical creativity; which seem to have no place in intuitionism. Not only that: Wilder saw this period of intuitionism “as as attempt to stem the flow of mathematical evolution — a kind of cultural resistance”. Most of this can be blamed on the then obsessive desire to root out “the threat of contradiction”. And, according to Wilder, that threat didn’t need “such drastic action as intuitionism demanded”.

Again, this seemed to be a sacrifice of speculation, conjecture and mathematical creativity on the part of intuitionism.

Despite these defects and criticisms, it was still the case — according to Wilder at least — that intuitionism “had a great and seemingly beneficial influence”. For example, a “number of prominent mathematicians shared in some, or all, of its tenets — for example, H. Poincaré and H. Weyl”.

To sum up.

Despite intuitionism’s radical nature and its rejection of the law of excluded middle (at least for infinite sets), Wilder nevertheless finished off by saying that

“its doctrine of constructivity was found to be adaptable to numerous situations within the framework of conventional mathematical theory”.

Completed Infinities

More technically, Wilder expressed his constructivist, anthropological and psychological position on mathematics by elaborating on the notion of a completed infinite. The idea of a completed infinite — at least initially — seems like a blatant and direct contradiction. How can any infinite set be completed (or complete)? If such a set is completed, then surely it’s not infinite.

So what, exactly, did Wilder say on the issue of the completed infinite? This:

“For example, an infinite decimal is not something that ‘just goes on and on without end’. It is to be conceived as a completed infinite, just as one conceived of the totality of natural numbers as a completed infinity.”

So Wilder actually gave us two examples of completed infinities — the “infinite decimal” and “the totality of natural numbers”. However, he didn’t really go into detail as to what such things actually are or what the words “completed infinite” mean. That said, the following words hint at an explanation (though they don’t help the non-mathematicians very much). Wilder wrote:

“Symbolically, it may be considered a second-order symbolism, in that it is not susceptible to complete perception, but is only conceptually perceivable.”

One may have a vague idea of what Wilder meant by the words directly above. Perhaps he was referring to a kind of “direct insight” (or intuition) into the nature of completed infinities. That is, completed infinities are “conceptually perceivable” (philosophers today would say conceivable); though they can’t literally — or even metaphorically — be seen.

Conclusion

From the retrospective point of 2021, the late-19th-century and early-20th-century obsession with the foundations of mathematics may seem strange. It may seem even stranger if we realise what the end result of this obsession was - at least according to Wilder - the following:

“The most powerful symbolic tools and his powers of abstraction and generalisation have failed the mathematicians in so far as ‘explaining’ what mathematics is, or in providing a secure ‘foundation’ and absolutely rigorous methods.”

It’s quite remarkable that Wilder claimed that modern mathematicians failed to explain what mathematics is considering the fact that even the layman would have a good go at the job. So there are two further questions:

Why can’t mathematicians explain what mathematics is? Why is this task so difficult?

Wilder argued that it was largely Kurt Gödel’s theorems which stopped mathematicians from “providing a secure foundation”; as well as from finding “absolutely rigorous methods”. Yet, from what Wilder wrote next, it seems as if mathematics not having any (secure) foundations (or not being free from all contradictions) might not have been such a bad thing. More precisely, Wilder went on to say that

“perfect rigour and absolute freedom from contradictions in mathematics are no more to be expected than are final and exact explanations of natural or social phenomena”.

Of course in science we don’t always — or ever — have “exact explanations of natural and social phenomena” and such things aren’t even expected. So is this really the case for mathematics as well? Perhaps this conclusion, on Wilder’s part, is simply a result of his constructivist and anthropological position on the practice and history of mathematics.

So to sum up with a single statement from Wilder.

It’s no surprise that Wilder’s general position was what it was if he believed that

“the only reality mathematical concepts have is as cultural elements or artefacts”.