[AntonioLao]

Reference: Morris Kline, ‘Mathematical Thought: From Ancient to Modern Times’, Volumes 1, 2, and 3, Oxford University Press, Oxford, 1972.

The mathematical foundation built upon three pillars of mathematical philosophy and as far as we know, reached an impasse and could possibly remain so for a very long time.

The problem started when mathematicians tried to axiomatize set theory in order to solve the paradoxes (contradictions) and the consistency problem.

The first foundational pillar is the school of logic founded by Russell and Whitehead.

The second foundational pillar is the school of intuitionist philosophy founded by Kronecker before the 20th century, modernized, and systematized by Brouwer beginning in 1907.

The third foundational pillar is the school of formalist philosophy founded by Hilbert.

So far, none of these philosophies achieved the objective of providing a universally accepted approach to mathematics.

[Guille]

I never herd about the second pillar, but I've been developing some philosophical thoughts on mathematics, which I want to give.

The basis of mathematics is logic. Logic is based upon a union between intuition and rational thinking. Logic is human created. Humans are subjective an so are everything that humans invent/create. Thus, logic is subjective. Logic has errors, failures, contradictions, fallacies... Which are, not only the evidence that logic is subjective but also that anything derived or based upon it, also is subjective (i.e. mathematics, science, human mind...).

I have developed some logic which proof several things about logic it self (like the fact that logic can't study philosophy as a whole). I will make other posts in the future in this thread, giving them.

I think the most important question to ask in philosophy of mathematics is:

Can mathematics study the whole of nature? How? Why/why not? Does, anyway, mathematics study nature?

I will post in these days about these questions.

[michellemfry]

I know about paradox's based upon words. Do you know of an example of a purely numerical paradox? I'm sure I'm asking the right person. I just can't think of one right now. I hope I'm not ignoring the obvious again.

[AntonioLao]

Originally Posted by michellemfry Do you know of an example of a purely numerical paradox?

A number that represents all the numbers? My wild guess for this would be infinity.

[Eric]

0 & 1 is the numerical paradox.

Infinity is the cause of the paradox.

[AntonioLao]

Originally Posted by Eric Infinity is the cause of the paradox.

For me, it's zero that is the paradox. If in the beginning was zero then nothing will ever come to be. On the other hand, zero could just mean that two repulsive primary forces are in perfect equilibrium.

[Eric]

Originally Posted by AntonioLao For me, it's zero that is the paradox. If in the beginning was zero then nothing will ever come to be. On the other hand, zero could just mean that two repulsive primary forces are in perfect equilibrium.

The paradox of both 0 & 1 that I referred to, are those that already exist in mathematics, and are a paradox because of infinity.

Now, the 0 you're refering to, is a case of a priori. This is not something that mathematics has dealt with. It is NOT the same 0. It requires setting aside a finite 0 & 1, and infinity; to think about a priori.

If in the "beginning" there was 0, then there would be no 1. What does this mean in plain English? If there was absolutely nothing there could never be anything else.

If absolutely nothing EXISTED, however, there wouldn't be absolutely nothing. Absolutely nothing is impossible. There is no such thing. It doesn't exist. It is a paradox of our own making.

There is an absolute 1. That is all there has ever been. There is no absolute 0 equal to or greater than it.

There is a 0 that is greater than a finite 0 or 1. But I'll stop here.

I guarantee you will keep going around in circles, by trying to think of it any other way.

[Lloyd Gillespie]

Originally Posted by Eric It is a paradox of our own making.

Then quit paradoxing it, Eric...

[Eric]

I already have.

[Lloyd Gillespie]

Originally Posted by Eric I already have.

Then state your logic in pragmatic terms, yet realize logic malfuncions when subjective mythologies are inferred into it___logic only functions properly when objective realities are inferred into it. Math is also an objective reality, founded on counting objects...

regards,