Monday, 22 February 2021

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.


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.


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”.


Wednesday, 17 February 2021

What Does the Question “What is it like to type these words?” Mean?


Here’s a passage from the David Chalmers’ well-known and important book The Conscious Mind: In Search of a Fundamental Theory. In this passage, the phrase “something it is is like” is used:

“We can say that a being is conscious if there is something it is like to be that being, to use a phrase made famous by Thomas Nagel. Similarly, a mental state is conscious if there is something it is like to be in that mental state. To put it another way, we can say that a mental state is conscious if it has a qualitative feel — an associated quality of experience. These qualitative feels are also known as phenomenal qualities, or qualia for short. The problem of explaining these phenomenal qualities is just the problem of explaining consciousness. This is the really hard part of the mind-body problem.”

The phrase “something it is like” is the main theme of this piece. I’ll tackle that by taking each sentence of the passage above, one at a time.

There’s Something it’s Like to Type These Words

Firstly, we have this opening sentence:

“We can say that a being is conscious if there is something it is like to be that being, to use a phrase made famous by Thomas Nagel.”

The American philosopher Thomas Nagel wrote a famous paper called ‘What is it Like to Be a Bat?’ in 1974. That paper inspired mountains of responses — both positive and negative. Nagel’s paper is widely thought to capture the main problem with any “physicalist” or scientific accounts of subjective experiences — whether those of bats or of human beings.

I can now ask readers the following question:

“What is it like to be you?”

And any reader can, in turn, ask me this question:

“What is it like to be you [i.e., me]?”

Indeed I can now ask myself this question:

“What was it like to be me yesterday or even ten minutes ago?”

I can even ask myself the following question:

“What is it like to type these words — right here and right now?”

So, on first glance, the words “something it is like” (or “what is it like to be…”) are hard to make concrete. That said, it’s of course the case that we intuitively seem to understand what it means. After all, isn’t there something it is like for me — right here and right now — to type these words? And isn’t there something it is like for you to read these words?

But what is this something it is like to type these words?

It’s definitely something very particular to me. And that, from the start, seems to make it an unscientific matter. (Perhaps non-scientific or even a-scientific would be a better term.) It may also be the case that the what it is like to type these words is incommunicable to other people (more of which later). I may of course attempt to do so in poetry, prose or in a subjective language. However, such descriptions (if that’s what they truly are) will never truly capture for someone else what it is like for me to to type these words. The best that can happen is that the person who reads (or hears) my accounts can place himself in some kind of “imaginative sympathy” with what it is that I’m describing. However, in that case it won’t be my words (or descriptions) alone which are truly capturing — for that other person — what it is like for me to type these words. This will basically mean that the hearer of my words will simply imagine himself typing his own words right here and right now. In that case, then, it won’t be my words alone which capture for him what it’s like for me to type these words.

Here’s another take on this.

Surely another person would literally need to be me in order to know what it’s this like for me to type these words. And that, surely, is impossible.

That said, let’s just accept that a sci-fi fusion is possible (if only for philosophical reasons). In other words, let’s accept that it’s logically possible that such a thing could happen. (See David Lewis’s ‘Survival and Identity’; which deals with both the fusion and fission of persons. Fission is when one person becomes two persons.)

Yet even if there could be some kind of fusion between myself and someone else, that fusion wouldn’t result in that other person experiencing what it’s like for me to type these words. Instead, it would result in this me-and-another-person fusion experiencing what it’s like to type these words. And even then, this person-fusion may still not be able to communicate “its” what-it-is-like-to-type-these-words experience to other people or even to itself.

What is it For a Mental State to be Conscious?

“Similarly, a mental state is conscious if there is something it is like to be in that mental state.”

Again, there’s something it is like to be in this mental state of typing these words. That is, I’m experiencing and feeling things right here and right now while writing these words.

What can be said about these experiences?

Nothing very precise or exact. Or, in the words of a physicist, I may be able to capture something qualitative; though not something quantitative. That said, perhaps nothing determinately qualitative can be captured either!

And it’s this very inability to communicate — or offer precise “verbal reports” — about this particular mental state (or sequence of states) which makes it suspect to philosophers like Daniel Dennett. However, the fact that these mental states (or experiences) can’t be adequately communicated — or communicated at all — doesn’t mean that these experiences aren’t real (or actual).

So does all that mean that the Dennetts of this world are demanding that at least something — or indeed everything — about this particular mental state of typing (right here right now) must be communicable? And if it isn’t, then is that the sole reason why such people reject — both from a scientific and a philosophical perspective — subjective (or “private”) mental states?

All the above means that the following is the bottom line:

If that something it is like to type these words is essentially incommunicable, then either it isn’t real or it serves no purpose in either science or philosophy. (Dennett can be read as arguing that x isn’t real precisely because it’s incommunicable and/or verifiable.)

Yet the feels I experience when typing these words clearly do exist (or are actual/real)… at least to me. That — again - is the problem. These feels only has an existence (or reality) to me and to me alone. And that’s why the Dennetts of this world have a problem with such things.

It’s All About Feels

“To put it another way, we can say that a mental state is conscious if it has a qualitative feel — an associated quality of experience.”

This seems to be suggesting — or stating — that a mental state must have a qualitative feel (or a set of qualitative feels) if it’s deemed to be conscious. In other words, can we make sense of a mental state without a qualitative feel?

More particularly, can we make sense of a mental state which involves, for example, mathematical calculations having no qualitative feels?

In this particular case it doesn’t matter if mathematics and equations have nothing to do with mental states — let alone with qualitative feels. This is about what happens when mathematical calculations are carried out by human beings.

Of course it can now be asked why anyone would care about the very particular — and (seemingly) subjective — “accompaniments” to mathematical calculations. However, that again doesn’t matter because the main argument here is that such subjective accompaniments to mathematical calculations can’t be described, grasped or owned by any science.

Nonetheless, what mathematicians say about what happens when they calculate is amenable to scientific description or study. That is, the verbal reports of mathematicians are out there in public space. Thus scientists and philosophers can indulge in heterophenomenological descriptions of the mathematical calculations which occur in human minds.

The word “heterophenomenological” has just be used.

The term heterophenomenology was coined by Dennett. He uses it to describe a third-person approach to the study of mental phenomena. This approach consists mainly in studying the verbal reports of subjects — such as mathematicians — and then making scientific and philosophical sense of them. And that is done in order to make mental states — such as mental calculations — scientifically acceptable. (See my ‘Against Daniel Dennett’s Heterophenomenology’.)

Meet Qualia

“These qualitative feels are also known as phenomenal qualities, or qualia for short.”

At first glance, saying that qualitative feels are phenomenal qualities doesn’t help us very much. We can now simply ask:

What are phenomenal qualities?

Well, they’re qualitative feels!

Now we can also say that both phenomenal qualities and qualitative feels are qualia:

So what are qualia?

Well, they’re phenomenal qualities and qualitative feels

Another problem here is that many accounts of qualia simply say what qualia aren’t, not what they are. And how could anyone truly say what qualia are anything other than that they’re phenomenal qualities and qualitative feels?

So here again we come up against a brick wall.

Can we describe such qualia/phenomenal feels/phenomenal qualities in a way that will take us anywhere that’s acceptable from a scientific point of view? More relevantly, can we say anything of philosophical substance?

Again, there is something it is like for me to type these words right here and right now. And, on that, I’m with the qualiaists. However, what can I say about that something? Nothing much — except in the guise of my own verbal reports. Yet, arguably, such reports don’t truly capture the supposedly private qualia I’m reporting. And, on that, I’m with Dennett in that he argues that there isn’t something “behind” the reports which can count in science or even in philosophy.

Explaining Qualia

“The problem of explaining these phenomenal qualities is just the problem of explaining consciousness. This is the really hard part of the mind-body problem.”

Firstly, David Chalmers assumes that there is what he calls a “hard problem”. There may well be; though that should never be simply assumed.

Secondly, what does Chalmers mean by the the word “explaining” (as in “explaining consciousness”)? That is, what is it to explain qualia or to explain consciousness?

What would an explanation of phenomenal qualities look like — even if there is one? This isn’t to say that there is — or there isn’t — an explanation: it’s to ask what such an explanation would look like. More particularly, I suspect that no explanation would ever satisfy David Chalmers… or Thomas Nagel for that matter. And that could quite possibly be because there can be no explanation which pleases everyone— at least not of the kind that Chalmers demands. Perhaps this is, after all, a bogus problem in that either it isn’t a problem at all; or because no explanation would ever satisfy those philosophers who’re demanding an explanation. (See my David Chalmers’ Unanswerable “Hard Question” About Consciousness | by Paul Austin Murphy | Predict | Medium.)

More technically, would such an explanation of phenomenal properties be an explanation from a first-person/subjective/phenomenological point of view? Well, that wouldn’t satisfy most scientists and many philosophers.

Okay. Would such an explanation be a neuroscientific/behaviourist/functionalist explanation? Well, that wouldn’t satisfy Chalmers, Nagel and many other philosophers.

So what about uniting the physical/functional accounts with first-person accounts? Is that even possible? Yes; in the particular respect that a person may verbally report his experience/s of particular phenomenal properties and that may satisfy Dennett and others. However, such reports would still be solely verbal reports of qualia — not accounts of qualia themselves (whatever that may mean). Indeed what would an account of a single quale even look like? And perhaps even if there were such an account, then it still couldn’t — almost by definition — be united with a scientific account.

Thus there’s a definitional gap and an “explanatory gap” between scientific accounts of phenomenal properties and subjective/first-person/phenomenological accounts of the (supposedly) very same things — and never the twain shall meet.

The upshot here is that David Chalmers will never be satisfied with the accounts of Daniel Dennett and (many) scientists. And Dennett and these scientists will never be satisfied with the accounts of Chalmers and the “mysterians”. Thus, again, there’s a chasm between the two positions. And even those much-advocated “structural correlations” (i.e., between neural states and conscious states) will certainly never bridge that chasm. Perhaps nothing will.


Note: A Single Mental State and its Set of Qualia?

It’s worth questioning if there is such a thing as a perfectly circumscribed and delineated single mental state. Don’t mental states “flow” into each other? And if mental states do indeed flow into each other, then there may not be single mental states at all.

Similarly, how many qualia make up a single mental state? Does that question make sense? Can it be answered? What would an answer look like? And even if a single mental state had a determinate number of qualia, how would we know what that number is?

So it seems that the full nature of “qualia” can’t be expressed or described. And that, again, is why the Dennetts of this world have a problem with such things.

Sunday, 7 February 2021

The Obsession With Essence and Necessity Began With Aristotle

The ancient and historically important philosophical notions of essence and necessity were vital to Aristotle’s overall philosophy. And, ever since Aristotle’s day, these philosophical notions have played a vital role in Western philosophy, theology, science and even in political theory.

It’s commonly agreed that the notion of essence began with Aristotle. However, a philosophical prototype can also be found in, for example, Plato’s Euthyphro. In that book Plato argued that physical entities acquire their essential being when they instantiate (or exemplify) what he called Forms. For Plato this meant that Forms are abstract universals which exist before any (concrete) particulars instantiate them. Plato therefore saw Forms as the paradigms of the particulars (i.e., things) which we experience in everyday life.

Aristotle himself departed from Plato in that he attempted to discover the (non-capitalised) form (or nucleus) of an individual physical entity. (Aristotle called such a thing ousia or substance.) In other words, Aristotle believed that Forms (or universals) must be instantiated in order to have being.

In more detail.

Aristotle used the Greek phrase τὸ τί ἦν εἶναι; which means “the what it was [or is] to be”. (The equivalent scholastic term is quiddity.) He also used the shorter phrase τὸ τί ἐστι; which means “the what it is”. (This is equivalent to the scholastic term haecceity). Then, in turn, these Greek phrases were rendered into the Latin term essentia (i.e., “essence”).

More relevantly to the following piece: Aristotle believed that gaining knowledge about the world is essentially about discovering what is essential and necessary to any given x.

The Purpose and Natural Place of Flames

Firstly, let’s take the often-cited example of Aristotle’s explanation of why fire (or flames) goes upwards. This is Aristotle on the subject:
“The ‘simple’ bodies, since they are four, fall into two pairs which belong to the two regions, each to each: for Fire and Air are forms of the body moving towards the ‘limit,’ while Earth and Water are forms of the body which moves towards the ‘centre’.”
Sure, it’s easy to be (retrospectively) smug about Aristotle’s views on fire and on much else. (Thus displaying a kind of Whig history of philosophy.) But, at the time, he had no reason not to conclude that flames displayed purposive behaviour. In addition, Aristotle relied on observations to come to his conclusions. After all, all flames do indeed move upwards. So, unlike Plato, Aristotle did pay much attention to the transitory (physical) things around him.

So why did Aristotle believe that flames go upward?

Aristotle reasoned in terms of what he called first principles. In the particular case of fire (or flames), he believed that fire seeks it “natural place” above the earth. This also meant that Aristotle believed that flames must necessarily rise upward.

Aristotle also believed he could discover was the “final cause” of things. In the case of flames, that final cause is the end of their upward journey. More technically, Aristotle argued that the four elements rise or fall toward to their natural place. (More on what Aristotle meant by the words “natural place” can be found here.) They do so in the concentric layers which surround the center of the earth and which form the sublunary spheres. More relevantly, the natural place of fire is higher than that of air but below the innermost celestial sphere (which contains the Moon).

This is, of course, Aristotle’s teleological account of things. Thus the telos of flames is to rise upward toward the heavens. In that sense, flames display “purposive behaviour” — their purpose is to rise to the heavens.

Necessity and Essence

As just hinted at, necessity is built into many of Aristotle’s explanations.

In basic terms, Aristotle believed that given causes necessitate given effects.

So let’s take flames (or fire) again.

Aristotle would have argued that the flames under a pan necessarily boil the water in the pan. In contemporary terms, this relation between flames and the boiling water in a pan is one of causal determination.

To return to the opening theme of flames rising upward.

So why is it also necessary that flames flow upward? Aristotle believed that this is because the essence of flames is that they flow upward. More abstractly:

Is it necessary that x displays behaviour (or action) A because x instantiates essence E.

Of course there’s a danger of conflating (or confusing) essence and necessity here. Surely only a physical (or perhaps abstract) property can be essential to any given x, not a specific behaviour (or action) of x.

However, why can’t a specific behaviour (or action) be seen as a property — even an essential property — of x? This means that the meaning of the word “essence” is — at least partly — a definitional (or stipulational) matter. That is, the word “essence” doesn’t itself have an essential meaning (or definition).

So we can conclude by arguing the following position:

If any given x always behaves (or acts) in a specific way, then that behaviour (or action) can be seen as being essential to x.

We can also link essence to behaviour (i.e., rather than seeing a specific behaviour — or action — as itself being essential). In this sense, we can argue that x’s essence E (i.e., a single property or a set of properties) brings about behavior A.

All this can be summed up and and stated in the following way:

If x has essence E, then it must behave (or act) in way A.

Yet even if there is essence E of x, how does E itself bring about behaviour A? In other words, how does E necessitate A?

The question as to whether behaviour can be essential is also relevant when it comes to subatomic particles. The following discussion on particles also raises the point that the philosophical notion of essence can be jettisoned entirely.

Relationalism: Particles and Mass

Subatomic particles are almost entirely defined in terms of their “relational” properties: such as their interactions with fields, forces or with other particles. (See my ‘Carlo Rovelli’s Relational Quantum Mechanics’.) Even the mass and charge of particles can’t be defined as (what philosophers call) intrinsic properties in that they’re determined by each particle’s place within a quantum system (or systems). More technically, a particle’s mass is determined by its relation to fields, forces and to other particles. And charge is similarly relational (see here).

This is the philosopher James Ladyman on this subject. He writes:

“[A] particle that never interacts with anything else could [never] have any value whatever for its mass.”


“since real particles will always interact with something or other [we can] ignore this”.

It can also be said that mass is “defined operationally” in that “the ratio of the masses of two particles is a constant of proportionality”. This too is a broader way of putting the better-known example which states that the number of electrons in an atom is equal to the number of protons. (Technically, a proton has a positive charge equal in magnitude to a unit of electron charge.)

Interestingly enough, Ladyman takes a particle’s lack of a (non-relational) essence to have the consequence that it can’t be what philosophers call an “individual”. (In broad terms, an individual is any given x which is deemed to be — to a large degree at least — self-sufficient, determinate and circumscribed.)

But firstly we need to see what this position is a reaction against. It’s largely a reaction against Leibniz. This is how Leibniz expressed his position on an individual

“The nature of an individual substance or of a complete being is to have a notion so complete that it is sufficient to contain and to allow us to deduce from it all the predicates of the subject to which this notion is attributed.”

On the other hand, James Ladyman and Don Ross — in relation to the physicist David Bohm’s position - say that

“[s]ince none of the physical properties ascribed to the particle will actually inhere in points of the trajectory, giving content to the claim that there is actually a ‘particle’ there would seem to require some notion of the raw stuff of the particle; in other words haecceities seem to be needed for the individuality of particles of Bohm theory too”.

The Irish philosopher Ernan McMullin also elaborated on this overall “relationist” position. He wrote:

“The use of namelike terms, such as ‘electron’, and the apparent causal simplicity of oil-drop or cloud-track experiments, could easily mislead one into supposing that electrons are very small localized individual entities with the standard mechanical properties of mass and momentum. Yet a bound electron might more accurately be thought of as a state of the system in which it is is bound than a separate discriminable entity… What is meant by ‘particle’ in this instance reduces to the expression of a force characteristic of a particular field[].”

Intuitive Knowledge of Essence and Necessity

We can now ask the following question:

How did Aristotle know about any given x’s essence or what is necessary about x’s behaviour?

An essence can’t be observed. And even if what is taken to be an essence could be observed, then that mere observation couldn’t tell a philosopher that such a property is essential. Properties don’t (as it were) broadcast their essential nature.

This meant that Aristotle couldn’t extract anything essential or necessary from flames (or any other given x) merely by observing them. Yet a similarly thing can be said about Isaac Newton and his discovery of the laws of gravity. In this case, too, Newton needed more than his apocryphal observation of a falling apple to conclude that it is gravity which accounts for such a thing. Newton also needed mathematical and theoretical insight — none of which could have been drawn solely from his observations.

Of course Aristotle himself would never have claimed that essences can be observed. Instead, essences are known through intuition.

(It can be questioned whether the word “intuition” is an accurate translation of anything Aristotle actually wrote. The word itself comes from the Medieval Latin intuitio; which can be translated as: “a looking at, [an] immediate cognition”.)

In the limited respect of his commitment to intuition, Aristotle was a Platonist.

Aristotle believed that intuition allows us to know (or “see”) things directly. That is, intuition allows us to make philosophical conclusions which go beyond observation, data or inductive support. However, unlike Plato, observation was still part of the overall story for Aristotle.

But what exactly is (philosophical) intuition?

Firstly, let’s ask the following question:

If we have direct access to the essence of any given x, then what would show the (philosophical) intuitor that he is wrong (or right) about what he concludes about x?

Interestingly enough, Aristotle is at one with those contemporary mathematical Platonists who similarly stress intuition. (See my ‘Platonist Roger Penrose Sees Mathematical Truths’.) Mathematical Platonists also claim to intuit (or “see”) those mathematical truths which haven’t — as yet — been proved.

So now, in turn, we can ask such a mathematician this question:

How do you know that some mathematical statements are true when they haven’t been proven to be true?


What w/could tell you that you’ve made a mistake about a mathematical statement’s truth?

And now to move back to Aristotle:

What is it, exactly, to intuit the essence of any given x or to intuit the necessary causes (or behavior) of any given x?

It may now seem that Aristotle’s position can be seen as an example of “a priori theorising” or rationalist reasoning. Yet Aristotle’s approach could never have been completely rationalist because he did, after all, take pains to observe such things as flames and he meticulously noted how they behaved.


To finish: it’s now worthwhile noting that Aristotle’s positions on essence and necessity are opposed to what occurred during — and after — what came to be called “the scientific revolution”.

More clearly, most scientists came to disregard the search for necessary truths completely. In terms of essence, what became important was not essence; but the motions of objects and the particles which constitute them. In addition, qualitative (or subjective) descriptions were substituted with quantitative descriptions. The search for “final causes” was also largely abandoned and scientists concentrated instead on finding (still to use Aristotelian terms) “efficient” material causes. Finally, although Aristotle relied to some degree on observations (as stated above), he didn’t carry out any experiments.