However, one should remember that mathematics is now considered as a creative human invention (cf. Wittgenstein), like music, literature, art and mathematicians can have “mathematistic license”.

This is a tendentious assertion, and false if you’re talking about the majority of practicing logicians and mathematicians (again, in my experience). Wittgenstein, who you have quoted time and again in this thread, was not a mathematician, and AFAIK made no major contributions to mathematical logic. His Tractatus Logico-philosophicus was not a work of logic but a work of philosophy of language, in the logical positivist vein. He was a philosopher of language.

Further, there are any number of problematic corollaries to the assertion that mathematics is created by humans. I view it as just as absurd as the claim that laws of nature are human inventions. That is to say, it is trivial or false: trivial in the sense that humans did come up with the symbols, but false in the sense that they would have been just as true had there never been any humans at all.

kkwan - 02 February 2010 07:33 AM

OTOH, pure mathematical proofs are getting so long and complex that eventually only computers can complete them.

By “are getting” you mean that some of them have been. In general they are not. There are an infinite number of possible mathematical proofs, and unless you can demonstrate somehow that the short ones are running out (which you cannot do), there is no reason to accept this assertion.

[ Edited: 02 February 2010 03:03 PM by dougsmith ]

To me, infinity (in mathematics) means the sum total (either by counting or by the sum product.) of all numbers and all sets and subsets of numbers, regardless if they are “countable” or “uncountable”. Comparing individual sets of numbers may yield a difference in count, but the count of any set can never exceed the total count of all the numbers and sets and its value can never exceed the total value of infinity.

Not sure what this means. But if you mean that the infinity of the reals is the same size as the infinity of the integers, then this has been proven wrong and is necessarily false.

infinity: 3. A quantity greater than any assignable quantity of the same kind.
“Comparing individual sets of numbers may yield a difference in count, but the count of any set can never exceed the total count of all the numbers and sets and its value can never exceed the total value of infinity.”

Sorry, the subject is well beyond my ability to make any assertions about its usefulness and the comment was gratuitous.

However, one should remember that mathematics is now considered as a creative human invention (cf. Wittgenstein), like music, literature, art and mathematicians can have “mathematistic license”.

This is a tendentious assertion, and false if you’re talking about the majority of practicing logicians and mathematicians (again, in my experience). Wittgenstein, who you have quoted time and again in this thread, was not a mathematician, and AFAIK made no major contributions to mathematical logic. His Tractatus Logico-philosophicus was not a work of logic but a work of philosophy of language, in the logical positivist vein. He was a philosopher of language.

Further, there are any number of problematic corollaries to the assertion that mathematics is created by humans. I view it as just as absurd as the claim that laws of nature are human inventions. That is to say, it is trivial or false: trivial in the sense that humans did come up with the symbols, but false in the sense that they would have been just as true had there never been any humans at all.

kkwan - 02 February 2010 07:33 AM

OTOH, pure mathematical proofs are getting so long and complex that eventually only computers can complete them.

By “are getting” you mean that some of them have been. In general they are not. There are an infinite number of possible mathematical proofs, and unless you can demonstrate somehow that the short ones are running out (which you cannot do), there is no reason to accept this assertion.

Ludwig Josef Johann Wittgenstein (26 April 1889 – 29 April 1951) was an Austrian-British philosopher who worked primarily in logic, the philosophy of mathematics, the philosophy of mind, and the philosophy of language.

Described by Bertrand Russell as “the most perfect example I have ever known of genius as traditionally conceived, passionate, profound, intense, and dominating,” Wittgenstein is considered by many to be the greatest philosopher of the 20th century.

To be dismissive of Wittgenstein as only a philosopher of language, not a productive logician/mathematician etc., is hubris. Must one be a logician/mathematician to comment on logic/mathematics? Are you one?

Since 1970 two other crises have arisen in mathematics, neither of which was anticipated, just as Godel’s work had not been. Both involve the issue of complexity: proofs that are too long and complex for anyone to be able to assert with total confidence that the theorems claimed are certainly true.

At first sight it seems obvious that the ‘crises of complexity’ that we will describe are epistemological in character, and say nothing about the ontology of mathematics. On the other hand some mathematicians prefer to think of mathematics as involving a process of creation rather than discovery, just as in architecture.

Just to be clear kkwan, you are welcome to use whoever you like as your authority for your views on philosophical topics. My point with Wittgenstein is simply that he is not an example of a logician nor of a mathematician, and his views are not, in my experience, in the mainstream of either math or logic.

The sort of argument I have just presented is ad hominem, however your tendency on this Forum has always been to quote other sources at length for your views. Using other sources rather than your own capabilities to string together an argument is really only a good strategy when those sources are authoritative and mainstream. Wittgenstein’s views do have some adherents within philosophical circles (though his logical positivist approach is now passé, in fact); however he is not an authority within logic or math. If you want to do some reading on the fundamentals of logic, there are many, many sources you can choose from. Wittgenstein is not one of them. That does not mean that his views about the metaphysics of mathematics are necessarily false, only that we have no particular reason to believe him an authority.

Given that he is not an authority, it’s probably better for you just to put the arguments into your own voice rather than quoting him instead, and pretending that we should believe what he says because of his name.

Cantor, Frege and Gödel, on the other hand ...

[ Edited: 02 February 2010 08:29 PM by dougsmith ]

Just to be clear kkwan, you are welcome to use whoever you like as your authority for your views on philosophical topics. My point with Wittgenstein is simply that he is not an example of a logician nor of a mathematician, and his views are not, in my experience, in the mainstream of either math or logic.

The sort of argument I have just presented is ad hominem, however your tendency on this Forum has always been to quote other sources at length for your views. Using other sources rather than your own capabilities to string together an argument is really only a good strategy when those sources are authoritative and mainstream. Wittgenstein’s views do have some adherents within philosophical circles (though his logical positivist approach is now passé, in fact); however he is not an authority within logic or math. If you want to do some reading on the fundamentals of logic, there are many, many sources you can choose from. Wittgenstein is not one of them. That does not mean that his views about the metaphysics of mathematics are necessarily false, only that we have no particular reason to believe him an authority.

Given that he is not an authority, it’s probably better for you just to put the arguments into your own voice rather than quoting him instead, and pretending that we should believe what he says because of his name.

Cantor, Frege and Gödel, on the other hand ...

To be fair to Wittgenstein, even though he was instrumental in leading philosophy into the “linguistic turn”, his books are notoriously obtuse, he was extremely eccentric and sprouted aphorisms like an oracle, nevertheless his philosophy is much more than the logical positivist approach. His philosophical investigations were far and wide. Here is another essay on Wittgenstein

Considered by some to be the greatest philosopher of the 20th century, Ludwig Wittgenstein played a central, if controversial, role in 20th-century analytic philosophy. He continues to influence current philosophical thought in topics as diverse as logic and language, perception and intention, ethics and religion, aesthetics and culture.

As you admit, there is a tad of ad hominem in your assessment of him. Anyway, enough of Wittgenstein. It is not essential to consider his philosophy exclusively. There are many other interesting philosophers, thinkers, logicians and mathematicians (living or deceased) well worth exploring.

To be fair to Wittgenstein, even though he was instrumental in leading philosophy into the “linguistic turn”, his books are notoriously obtuse, he was extremely eccentric and sprouted aphorisms like an oracle, nevertheless his philosophy is much more than the logical positivist approach. His philosophical investigations were far and wide. Here is another essay on Wittgenstein

Yes, I know; FWIW I spent an entire semester at university studying his Investigations. It’s his previous book (the Tractatus) that has the logical positivism.

IMO he was brilliant at asking interesting questions, but highly overrated in the public imagination. He was basically something of a philosophical gadfly. (Unless you are a logical positivist, of course, and then you’ll particularly enjoy the Tractatus).

Why is the concept of the infinite so fascinating and so paradoxical? Is it because as humans, we are limited by life and intellect, but have no reason to think Nature has such restrictions?

Science has revealed to us the possibility of the universe (in the realm of the very large) and the underlying reality of vagueness (in the quantum realm of the very small), is infinite.

The notions of a beginning and an end are necessary in finite processes. However, for infinite processes, these notions make no sense. The completed infinity is a contradiction-in-terms if infinite processes have no beginning or end.

Some theories predict that the Universe doesn’t have a beginning at all, but that if you follow it backward in time, it eventually bounces, almost like a ball, into a previous state in which it was contracting. The Universe may behave cyclically — contracting, expanding and contracting again — or it may be that it bounced into expansion only once and will keep on expanding forever. Another possibility is that the Universe began in some rather uninteresting stationary state, and then started to expand due to the effect of quantum fluctuations. In that scenario, the expansion has a beginning, but the Universe itself doesn’t necessarily have one.

And the bubble multiverse:

One rather shocking thing about this bubble making is that the whole process need not have a beginning or an end.