Raymond L. Wilder on the Foundations of Mathematics

From the outside, the 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. According to Raymond L. Wilder, the modern mathematicians with


"his 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". (197)


It's quite remarkable that Wilder claims that the modern mathematician has failed to explain what mathematics is considering the fact that even the layman would have a good go at the job.


The question is: Why can’t they explain what mathematics is?


Why is this task so difficult?


Was it Gödel’s results that stopped mathematics from ‘providing a secure foundation’ as well as ‘absolutely rigorous methods’? Is it really the case that the search for foundations, as well as for absolutely rigorous methods, is well and truly over, let alone when Wilder wrote these words in 1968?


From what Wilder says next, it seems as if mathematics not having any foundations, or not being free from all contradictions, may not be such a bad thing. More precisely, he writes 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". (197)


And, of course, in science we don't have "exact explanations of natural and social phenomena" and nor are such things ‘expected’ in science. Is this really the case in mathematics as well? Surely not. Perhaps this conclusion, on Wilder’s part, is simply a result of his materialist, sociological or even Marxist position on the practice and history of mathematics. Surely even these positions accept different standards from maths – indeed, they do.


Again, it is no surprise that Wilder says what he says if he that "the only reality mathematical concepts have is as cultural elements or artefacts" (197). This position seems to go even further than constructivism; though perhaps not as far as the late Wittgenstein.


More technically, Wilder expresses his constructivist, Marxist or sociological position on mathematics by elaborating on the notion of a ‘completed infinite’ (198). This sounds like a blatant and direct contradiction. How can any infinite be complete or completed? If it is completed, then surely it's not infinite. What, exactly, does Wilder say on this issue of the completed infinite? -


"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." (198)


Wilder gives us examples of completed infinities, the infinite decimal and ‘the totality of natural numbers’; though he doesn’t say what such things actually are or what the phrase ‘completed infinite’ means. The following hints at an explanation; though it doesn't help the non-mathematicians much. He writes:


"Symbolically, it may be considered a second-order symbolism, in that it is not susceptible to complete perception, but is only conceptually perceivable." (198)


Do you have a vague idea of what Wilder means by the above? Perhaps it's a kind of ‘direct insight’ or intuition into the nature of completed infinities. It can be conceptually perceived; though not seen – literally or even non-literally.

 

Gödel's Incompleteness Theorem & Leibniz’s Dream

Albert Einstein and Kurt Godel

The mathematician and educator, Morris Kline, once made a rather grand claim about Kurt Gödel’s Incompleteness Theorems when he (in his Mathematics: The Loss of Certainty) said that it

"was a response to Leibniz’s 250-year-old dream of finding a system of logic powerful enough to calculate questions of law, politics, and ethics".

Perhaps Leibniz’s dream had nothing to do with applying logic to the content of law, politics and ethics; but only to the form of the arguments in which these things were expressed. For example, in ethics, logic can't show us “what is good”. However, it can detect good and bad arguments as to what constitutes “the Good”.

Similarly logic can show faulty reasoning in political and legal debate; regardless of the actual content of these debates.

So, in that sense, it's indeed true that logic can be applied to law, politics and ethics – indeed to anything! So just as the premises of a deductive argument needn't be true in order for the argument to be valid; so the content of political, legal and ethical statements doesn't matter to the logician - though what follows from them, logically, does matter him.

Logic can "provide the tools to resolve ethical questions by mere calculation" if it dealt only with form and not with metaphysical, epistemological and semantic content.

In any case, were Gödel’s theorems really a response to Leibniz’s dream? Perhaps it was just Gödel’s way of showing us that, well, an axiomatic system (or mathematics generally) can't be both fully consistent and complete – that’s it (without philosophical knobs on).

Much has been made of Gödel’s theorem by non-mathematicians and by many non-philosophers. Morris Kline expresses much of this here. He writes that we

"might think that Gödel’s proof implies that the rational mind is limited in its ability to understand the universe".

How a result in metamathematics could do that (even in principle), I’m not sure. In any case, the mind, again in principle, must surely be limited in some way or ways. Perhaps that means that it could never understand everything there is to know about an infinite universe. Indeed this is bound to be the case. Only an omniscient mind could know everything there is to know about the universe.

Kline makes this point. He says that

"though the mind may have its limitations, Gödel’s result doesn’t prove that these limitations exist".

What is limited isn't the mind as such; but that "axiomatic systems are limited in how well they can be used to model other types of phenomena". This has nothing to do with the mind of man taken generically! It's to do with axiomatic systems and the modelling of other types of phenomena.

Not only that: the "mind may possess far greater capacities than an axiomatic system or a Turing machine". I would say that of course the mind does actually possess far greater capacities than an axiomatic system or a Turing machine. Evidently! For a start, the mind can create great poems or pieces of music. It has memory, experience, imagination, the ability to dream, create, invent, manipulate the environment and so on. Some of these things Turing machines can do; though many of them they can’t do. And no single axiomatic system or Turning machine can do all the things a human mind can do – not even a deranged or damaged human mind!

Another common supposed result of Gödel’s theorems is to assume that his proof implies a limit to artificial intelligence. Perhaps this is a more feasible idea because it must be about the mathematical limitations of artificial intelligence – and that would be relevant to Gödel’s proof. That is, would an indefinite advance in AI be halted by the result of Gödel’s proof which showed that if a mathematical system (therefore all combined) can't be both complete and fully consistent, then a project that relies on mathematics (that is, AI) will never be both complete and fully consistent? Thus there will be a limit to what AI can do.