Logic and Language

Logic and Language

Copyright © James R Meyer 2012 - 2016 www.jamesrmeyer.com

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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.

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”*.

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 - then you 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.

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, and absurd concept and has now been comprehensively shown to be incorrect.

This attachment to a completely unfounded belief in Platonism has severely held back progress in mathematics and logic. It is imperative that this impediment to progress is now removed.

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 it 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 perusing 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’*.

Logicians will tell you that Tarski defined truth. (Footnote: See, for example, the entry Tarski’s Truth Definitions in the Stanford Encyclopedia.) Most statements we encounter in ordinary language refer to the real world, and to a finite number of things. We generally accept, at least that such a statement is true or false, and when we say that, we are effectively saying that it is decidable, by a finite amount of examination of the real world, if the statement is correct or not. Certainly, in many cases, we would not actually do that examination, but we accept that, given sufficient resources, such an examination is possible.

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

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”*

is provable.

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.

Why?

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

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?

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.

Footnotes:

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There is a new addition to the page Yet another flawed incompleteness proof, where Berto’s proof of incompleteness in his book *There’s something about Gödel* comes under scrutiny.

I found that making, adding or deleting footnotes in the traditional manner proved to be a major pain. So I developed a different system for footnotes which makes inserting or changing footnotes a doddle. You can check it out at Easy Footnotes for Web Pages (Accessibility friendly).

I have now added a new section to my paper on Russell O’Connor’s claim of a computer verified incompleteness proof. This shows that the flaw in the proof arises from a reliance on definitions that include unacceptable assumptions - assumptions that are not actually checked by the computer code. See also the new page Representability.

There is now a new page on Chaitin’s Constant (Chaitin’s Omega), which demonstrates that Chaitin has failed to prove that it is actually algorithmically irreducible.

Flawed proofs of the Diagonal Lemma by Panu Raatikainen and Vann McGee have been added to the Diagonal Lemma web page.

16th Mar 2015 Bishops Dancing with Pixies?

23rd Feb 2015 Artificial Intelligence

31 Mar 2015 Cranks and Crackpots

For convenience, there are now two pages on this site with links to various material relating to Gödel and the Incompleteness Theorem

– a page with general links:

– and a page relating specifically to the Gödel mind-machine debate:

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Copyright © James R Meyer 2012 - 2016

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