Page last updated 17 Nov 2020
What is Platonism? Exactly what Platonism should actually be defined as is a matter of some disagreement. It is probably fair to say that the ethos of Platonism as it stands today is quite far removed from that envisaged by Plato himself, after whom Platonism takes its name (see also The Myths of Platonism and Platonism’s Logical Blunder).
Plato believed that there exist perfect forms - forms which real physical things can never attain. So a triangle drawn in the sand can never be as exact as the notion of a perfect triangle with perfect lines and perfect corners.
Platonism, as it applies to mathematics and logic today, is the belief that mathematical concepts can be said to “exist” in some sense other than when described by some language. In other words, they do not need any language in order to “exist”, and as such, are considered as Platonists to be as real as physical objects.
So, mathematical objects such as ideal triangles, circles, and numbers are Platonic “entities”. Platonists have gone to extreme lengths to try to justify these beliefs, where almost any ridiculous argument is put forward to support Platonism, a good example being the Indispensability Argument.
No-one has ever shown that the concept that things referred to by rigorous mathematical and logical language can have some sort of independent existence has any usefulness whatsoever.
And, if one applies the principle of Occam’s razor - that one should only use as much as is needed to explain anything - there is no reason for anyone to start off with any assumptions that the mathematical symbols that mathematicians use refer to real entities that actually exist in some non-physical sense.
The contradictions inherent in Platonism
But the most damming argument against Platonism in mathematics is that it leads directly to contradictions; this is easily demonstrated, see for example Sums of infinitely many fractions 1, Sums of infinitely many fractions 2 and The Courant & Robbins Contradiction.
Despite this, most mathematicians, logicians and philosophers today appear to be Platonists. Much of the reason for this can be laid at the door of Gödel’s proof of his incompleteness theorem, a proof which purportedly showed that formal language could never prove ‘truths’ that could be demonstrated to be true in natural language. This is, and always was, an absurd concept and has now been comprehensively shown to be incorrect, see The flaw in Gödel’s incompleteness proof.
Georg Cantor also was an ardent Platonist, and his beliefs led to the notion that there could be a limitlessness that could be limited to be smaller than another limitlessness; you can read online here an English translation of Cantor’s work Grundlagen einer allgemeinen Mannigfaltigkeitslehre (Foundations of a general theory of sets), which lays out his philosophy on different sizes of infinity.
It is ironic and depressing that in this day and age, mathematicians and logicians can believe in the “real” “existence” of such contradictory things - and believe that these things have specific properties. But it is forever impossible to prove either that these things “exist”, or that they actually have the properties they are claimed to have.
In the field of mathematics, a fundamental tenet is that your pronouncements should all either be axioms, or statements that you prove from those axioms. And there is no physical proof, and there cannot be any proof, that there “exist” invisible non-physical “things” that do not interact with the physical world in any way, but which happen in some way to correspond precisely to certain sequences of symbols in a mathematical statement. Which leaves the option of simply making axiomatic assumptions, that have no logical foundation, that there “exist” mathematical “things” independently of any definition. But because the properties of these imagined “things” cannot be proven, then the properties of these imagined “things” must be added by definitions. And that means that such Platonist axiomatic assumptions add absolutely nothing to mathematics - they are utterly pointless, since these definitions are what are used by mathematical proofs. Furthermore, assumptions such as a “sum of infinitely many values” are inherently contradictory. (Footnote: See Sums of infinitely many fractions 1, Sums of infinitely many fractions 2 and The Courant & Robbins Contradiction.)
The attachment to a completely unfounded and unprovable belief in Platonism has severely held back progress in mathematics and logic. It is imperative that this impediment to progress is removed. See also Why do people believe weird things?
The page Shaughan Lavine’s “Understanding the Infinite” shows how an author who is so blinded by his Platonist beliefs that he engages in a willfully disgraceful misrepresentation of historical facts as well as numerous logical fallacies.
The notion of ‘truth’
It is sadly ironic that so much of what passes for mathematics, logic and philosophy is based on the absence of any definition of ‘truth’ which does not involve Platonism in some form. The notion of ‘truth’, along with some other notions has generated an amazing amount of confusion in the fields of mathematics, logic and philosophy.
In the past, a clear definition of ‘truth’ was avoided, fed by the implicit assumption that in some way, in some imagined Platonic reality, it had some sort of firm and unyielding basis. But when the notion of “provable” was discovered, it was simply assumed that “provable” and “true” were two entirely separate and distinct concepts. And at the same time, ‘truth’ was assumed to be something that had a distinct Platonic reality that was as definite as the notion of “provable”. With those assumptions came the associated notion that something might be “true”- even though it might never be possible to prove it - ever.
So, at a stroke, the concept of an assertion that could, in some unattainable sense, be “true”, but never provable, came into common acceptance. This, of course, was just one of the concepts envisaged by the Platonists. This concept has become so ingrained in the academic worlds of mathematics, logic and philosophy that at present, any suggestion that it is a concept that has no real utility and should therefore be dispensed with is met with derision.
Future generations will, no doubt, be amused by the current concept of mathematical ‘truth’, and view it as a rather quaint notion which has no place in their studies other than as an illustration of the folly of trying to peruse a logical analysis of that which eludes definition. Which brings to mind the aphorism: ‘A mathematician is a blind man in a dark room looking for a black cat which isn’t there’.
But, unlike most statements that refer to everyday experience, a mathematical statement can refer to infinitely many things, such as numbers. And we can have mathematical statements like:
For every mathematical entity x, x has the property P
where there are infinitely many x’s.
So the statement claims that every x (and there are an infinite number of them), has a certain property P. Tarski’s definition of ‘truth’ for the statement effectively is the proposition: (Footnote: Note that this is a simplification of Tarski’s definition; the simplification is to give the essence of the commonly accepted mathematical definition of truth without getting bogged down by details.)
The statement “For every mathematical entity x, x has the property P” is ‘true’ if every statement “every x has the property P” is provable (note that there are an infinite number of these statements).
Clearly, there is no way of going through the infinite number of mathematical entities (each x) one by one and checking if they have that property - because you can never finish that process. But, perhaps you could prove the statement from the axioms and rules of inference of whatever mathematical system you are working with? Now, just suppose for a moment that for this particular property, there is no way of deducing the ‘truth’ of the statement ‘For every mathematical entity x, x has the property P’ from the axioms and rules of whatever mathematical system you are working with.
According to Tarski, the original statement is ‘true’ if every possible individual sub-statement for that statement
“every x has the property P”
At first glance that appears innocent enough. Now, while if we take any one individual x, we may be able determine if x has the property P, this relies on a Platonist assumption, an assumption that is completely unacceptable to any person who thinks that mathematics should be based on science and logic rather than blind faith.
Because, now you are required to believe in the independent existence of an infinite number of these x’s, that a limitless quantities of such x’s actually exist, and have always existed in some sense in some non-physical ‘reality’, whatever that might mean - and that each such x either has or has not this property P. We are required to believe that these x’s, that we refer to with mathematical symbols, always existed before anyone even thought of mathematical symbols - and that each of these x’s always either had or had not the property P for all time. And this applies to all mathematical things, and all mathematical properties.
This commonly accepted mathematical notion of ‘truth’ does not allow you to reject such unfounded beliefs, It does not allow you to simply consider that combinations of mathematical symbols are simply an invention of the human mind, based on a few initial premises, and which follow a certain set of rules. Because if combinations of mathematical symbols are simply an invention of the human mind and which do not refer to real non-physical things (whatever that might mean), then a proposition like:
‘For every mathematical entity x, x has the property P’
does not have any ‘truth’ value that is independent of these initial premises and rules that we have for such mathematical statements.
Of course, if we discover a way of proving that statement, then we prove it from those premises and those rules - but we can never prove it by going through an infinite number of possibilities one by one.
For examples of how philosophers have managed to construct a huge edifice around the notion of ‘truth’ with reams of material based on the notion that mathematical truth is some sort of independently real ‘thing’, see the page Gödel and Machines, and also
PDF Deflationism and the Gödel Phenomena,
PDF Deflationism and the Gödel Phenomena: Reply to Cieślińsk,
PDF Deflationism and the Gödel Phenomena: Reply to Ketland,
PDF Does Gödel’s Incompleteness Theorem Prove That Truth Transcends Proof ?
Properties of non-physical ‘things’?
So why should anyone introduce an assumption which we can never, by the very nature of such an assumption, have any evidence to support it? Of course, anyone is entitled to have such beliefs if they so desire. But why should anyone agree to use such beliefs and such assumptions as a basis for all of mathematics? The fact is that much of what is called mathematics is based on Platonist beliefs - the assumption of the real but non-physical existence of mathematical things which have, and have always had, inalterable properties for all time.
It is interesting to look at the history of mathematics, and consider what happened when mathematicians discovered that their beliefs in some of these ‘real’ mathematical things led to contradictions. For example, when mathematicians discovered that the then current theory of sets led to contradictions, the response was, “There is nothing wrong with our Platonist viewpoint. There are real sets and they actually exist - it’s just that it’s our description of them that is at fault”.
This attitude leads to an endless chasing after the ‘real’ properties of the ‘real’ things that are believed to ‘exist’. What use is the assumption that there are ‘real’ mathematical things if you can never know what their properties actually are?
Ask a Platonist, “How do you know that your descriptions of what you consider to be the ‘real’ mathematical things are correct?” Of course, he can’t know - but even though he can’t know, he will insist that his description of mathematical ‘reality’ is the correct description ! He will insist that his version of mathematics is correct, while a non-Platonist mathematics, which uses less assumptions than his, is not.
Platonism as transcendental spirituality
An example of a certain type of Platonist mindset is exemplified by this extract: (Footnote: Richard Tieszen, After Gödel: Platonism and rationalism in mathematics and logic, Oxford University Press, 2011, Ch 4.)
“The defensible version of platonism I have in mind is what I called constituted platonism … According to constituted platonism, the transcendental ego as monad constitutes in a rationally motivated manner in classical mathematical practice the meaning of being of the objects toward which it is directed as ideal or abstract and non-mental. It constitutes such objects or concepts as transcendent or mind-independent. It is not possible to understand the constitution of meaning of being of such objects and, thus, constituted platonism, without seeing that human consciousness exhibits intentionality … Constituted platonism is what emerges if we develop the transcendental method in light of the intentionality of human consciousness in a way that allows us to accommodate the rational intuition of abstract concepts or objects. The way to bridge the gap between human subjectivity and mathematical objectivity is to fill in the account of the kinds of founded intentional acts and processes that make the constitution of the meaning of being of mathematical and logical objects possible.”
There is nothing to distinguish this sort of horrendous verbal slush from New Age mysticism - it is verbal garbage posturing as intellectual philosophy. That anyone would think that this sort of pretentious hokum could provide any kind of justification for Platonism to a non-Platonist is totally mind-boggling. It is, in a word, bullshit. (Footnote: Harry G Frankfurt: PDF On bullshit, originally published in the Raritan Quarterly Review in 1986, published as book by Princeton University Press, 2009. ) (Footnote: Gordon Pennycook et al: PDF On the reception and detection of pseudo-profound bullshit, Judgment and Decision making 10.6 (2015): 549-563. ) (Footnote: Gordon Pennycook et al: PDF It’s still bullshit: Reply to Dalton (2016), Judgment and Decision making 11.1 (2016): 123. ) (Footnote: Online bullshit generator )