First, I'm no trained mathematician. But I love the stuff! So that confession aside, I would love to talk about this. Generally, it seems there are attempts to found mathematics on logic, intuition, or to reduce it to matter of manipulating symbols according to rules (formalism). Correct me if I am wrong. I like all 3 approaches but feel that there must be a fundamental intuition of pure abstract unity/being. And that logicism and formalism can proceed from there. However I simply love the subject, and welcome opposing viewpoints. Let's talk about this great issue.

Hi Reconstructo,

I too am fascinated with the foundations of mathematics and logic.

I think its relavent to ask: Is there only one logical foundation of mathematics? I believe that it is true.

Can we develop a logic that includes the theorem that 1+1=3 ? I don't think so, do you?
That is to say, 1+1=2 is true in any logic that is capeable of asserting it.

The methods of logicism seem to me correct.
The results of Russell and Whitehead (Principia Mathematica) are correct (imo) as to the production of Analysis from predicate logic, (In spite of Godel's incompleteness theorems).

If you don't think so, what theorems of Principia would be rejected because of Godel?
Because there are undecidable propositions of Principia, does not entail that there are false theorems in that work.

I can't talk about the Principia. I have not read it. Nor am I against logicism. I suppose I do think there is a root intuition of abstract unity necessary for the symbols in that book or any other on the subject to be persuasive for us.

To even look at a mathematical or logical symbol as one symbol, with one meaning seems already to rely on this fundamental intuition. Do you feel that the natural numbers are self-evident? Is justification for these any more persuasive, in your opinion, than the intuition of them?

I feel like we have an intuition of unity, the one, and iterate this intuition systematically. Between all integers is the same "distance" of this one. I suppose for me logic and mathematics are both founded on this same something, which I have knicknamed the proto-concept. Now I have more philosophical than mathematical background by far. In the last few months I have been learning quite a bit, but it's only been a few months.

Aside from foundational questions, does math strike you as beautiful? Do you like e as if it were a sculpture? Or Euler's identity? Perhaps it's my aesthetic response to mathematics and logic that makes me feel it is innate. I'm quite interested in Kronecker. Also computer programming.

My approach to foundations is probably eccentric and focused on the basic intuition of unity and also the attempted intuition of infinity, potential or actual. What do you make of Cantor? I think he's fascinating, but I don't know if I can embrace all his infinities as real. Do you think we can think actual infinity? I thought Chaitin's book Metamath was great.

I am not convinced that 'self evidence' is sensible.
What proposition can show its own truth?
One is the set of unit sets, seems to work.

Eulers identity, is one among many interesting theorems concerning epsilon, pi and root (-1). See Hyperbolic: h-sin and h-cos and h-tan for more of that connection.

Kronecker..Is he the one who said: God provided the natural numbers and man provides the discovery of the resulting theorems?

Cantor with his heirarchy of infinities, is certainly fascinating.
Note, his definition and application of identity as a one to one correspondence allows this possibility, imo.
"Real infinities", infinities within physical reality seem very dubious.

We understand the notion of 'natural number without end' without being committed to a unique number 'infinity'.

I don't think that aesthetics or ethics has a logical foundation.

I can't talk about the Principia. I have not read it.

Nor am I against logicism.

I suppose I do think there is a root intuition of abstract unity necessary for the symbols in that book or any other on the subject to be persuasive for us. To even look at a mathematical or logical symbol as one symbol, with one meaning seems already to rely on this fundamental intuition.

Do you feel that the natural numbers are self-evident?

Is justification for these any more persuasive, in your opinion, than the intuition of them?

Aside from foundational questions, does math strike you as beautiful? Do you like e as if it were a sculpture? Or Euler's identity? Perhaps it's my aesthetic response to mathematics and logic that makes me feel it is innate.

First, I'm no trained mathematician. But I love the stuff! So that confession aside, I would love to talk about this. Generally, it seems there are attempts to found mathematics on logic, intuition, or to reduce it to matter of manipulating symbols according to rules (formalism). Correct me if I am wrong. I like all 3 approaches but feel that there must be a fundamental intuition of pure abstract unity/being. And that logicism and formalism can proceed from there. However I simply love the subject, and welcome opposing viewpoints. Let's talk about this great issue.

Don't worry: nobody else has, either! I'd be quite surprised to learn that even Godel read it right through, before finding the flaw in it.

My aesthetic appreciation of mathematics has always been patchy, at best. Thinking of the relationship in sexual terms: there has not been enough love in it, nor has there been enough pleasure. It's a crazy obsession, an infatuation, something I have even now not learned to sanely contain. I seem to need to repress it in order to cope with it at all. In almost every respect my obsession with mathematics parallels my ludicrous and unspeakably frustrating sex life, about which, however, I do not intend to go into detail!

The set of unit sets? Is that redundant? No offense intended. Frege wrote about how hard it was to define unity except by synonyms. And I agree that if unity is an intuition, it can't be proven. It would just be. I truly think it's an important issue. It's simplicity is perhaps deceptive. Of course I could be wrong.

Actually, didn't you mean "e" instead of epsilon? Are h-sin and h-cos the same as sinh and cosh? I've read about these but never used them. Interesting though.

Yes, Kronecker said pretty much that. "The rest is the work of man." He is coming back a little because of the nature of computers, which can only work algorithmically w/ the finite. As you probably already know.

I agree that real infinities are dubious. But I don't think we can even imagine real infinity. Or can we? I know he has his proofs. But it seems to me hard to tell one uncountable infinity from another. Doesn't he do this with power sets? Strange though.

What flaw are you referring to?
What theorems of Principia are false as a result of Godel?
I don't believe there are such flaws in PM.

Well, the first step is going to loving the subject to actually knowing the different positions. Bear in mind that new views are being advanced all the time.

I assume that Kronecker was quite religious, even about mathematics...."God did it". That kind of reasoning does not work for me, how about you?
How does the nature of computers relate to Kronecker's "God did it" reasoning?

It is interesting to note that, computers deal with logical calculations (~ v & -> <->) and mathematical calculations (+,-, *, /, ^ etc.) of bits and bites.
Machine language allows the question: what is the value of (2 or 3) just as easy as what is the value of (2*3).

When I was younger I wanted to produce an "Argumentor", that is, a program which could 'calculate' the value of any argument that involved a finite number of terms and propositions.
Do you think this project would be worthwhile?

Don't worry: nobody else has, either! I'd be quite surprised to learn that even Godel read it right through, before finding the flaw in it.

Cantor claims a set is infinite if a proper subset can have a 'one to one' relationship with it.
For example the number of even natural numbers can be 'one to one' with all of the natural numbers.
To show that there are more 'real' numbers than 'natural' numbers he invented the diagonal method.

No, it is not redundant. Russell shows how the notion of unit sets does not assume the number 1.

I'm pretty much inclined to go with something like logicism, but I don't yet know how to elaborate on the "something like".

Indeed (although I'm not quite sure how much this is related to what you are saying), the mathematics of formal systems, their discrete 'geometry' so to speak, self-evidently cannot itself be entirely formal.

Not at all, no.

I believe it should be much more persuasive.

My aesthetic appreciation of mathematics has always been patchy, at best. Thinking of the relationship in sexual terms: there has not been enough love in it, nor has there been enough pleasure. It's a crazy obsession, an infatuation, something I have even now not learned to sanely contain. I seem to need to repress it in order to cope with it at all. In almost every respect my obsession with mathematics parallels my ludicrous and unspeakably frustrating sex life, about which, however, I do not intend to go into detail!

Cantor claims a set is infinite if a proper subset can have a 'one to one' relationship with it.
For example the number of even natural numbers can be 'one to one' with all of the natural numbers.
To show that there are more 'real' numbers than 'natural' numbers he invented the diagonal method.

I am interested in the phrase "all of the natural numbers". It seems ambiguous to me: Cantor believed that there is an actual infinity of natural numbers, but a constructivist would presumably interpret "all" in a finite sense, e.g. "all the natural numbers that have so far been humanly conceived" or something like that. To use the phrase "all the natural numbers" in an attempt to prove that actual infinities exist would therefore beg the question.

I tend to think that the idea of actual infinity is incoherent, theoretically as well as physically. To me it seems like a conceptual conjuring trick - like a deck of cards that one is permitted to view end-on but not side-on. Before I believe in an infinite set, I want to be able (in principle) to count every individual "card" in the set - which is obviously impossible. From any conceivable point of view, there is a "now", and at every possible "now" an unlimited count will be incomplete. If that is so, Cantor's claims simply do not work - for example, the amount of natural numbers counted "up to now" will always be (exactly or approximately) twice the amount of even natural numbers counted up to this point.

But I would be interested to hear any counter-arguments.