On the Foundation(s) of Mathematics

1. Philosophy Forum
2. » Logic
3. » On the Foundation(s) of Mathematics

Sun 6 Jun, 2010 09:08 pm
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.

Owen phil

Mon 7 Jun, 2010 03:03 am
@Reconstructo,
Reconstructo;174050 wrote:
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.

Reconstructo

Mon 7 Jun, 2010 03:19 am
@Owen phil,
Owen;174132 wrote:
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.

Owen phil

Mon 7 Jun, 2010 04:00 am
@Reconstructo,
Reconstructo;174135 wrote:
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'.

The calculus had difficulty being logically justified when it was based on 'infinitesimals', but thanks to Cauchy, the limit function avoids them.

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

Reconstructo

Mon 7 Jun, 2010 04:16 am
@Owen phil,
Owen;174145 wrote:
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.

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.

---------- Post added 06-07-2010 at 05:19 AM ----------

Owen;174145 wrote:

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?

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.

---------- Post added 06-07-2010 at 05:22 AM ----------

Owen;174145 wrote:

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 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.

I agree that infinity is not a number. It's a strange sort of notion, isn't it? And yet quite valuable.

---------- Post added 06-07-2010 at 05:23 AM ----------

Owen;174145 wrote:

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

I agree. But do you find a certain beauty in logic? Does logic have an aesthetic or intuitional foundation? Or is it just definitions and the movement of symbols?

Twirlip

Mon 7 Jun, 2010 04:50 am
@Reconstructo,
Reconstructo;174135 wrote:

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.
Reconstructo;174135 wrote:
Nor am I against logicism.

I'm pretty much inclined to go with something like logicism, but I don't yet know how to elaborate on the "something like".
Reconstructo;174135 wrote:
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.

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.
Reconstructo;174135 wrote:
Do you feel that the natural numbers are self-evident?

Not at all, no.
Reconstructo;174135 wrote:
Is justification for these any more persuasive, in your opinion, than the intuition of them?

I believe it should be much more persuasive.
Reconstructo;174135 wrote:
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.

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!

TuringEquivalent

Mon 7 Jun, 2010 06:32 am
@Reconstructo,
Reconstructo;174050 wrote:
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.

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.

Owen phil

Mon 7 Jun, 2010 07:00 am
@Twirlip,
Twirlip;174158 wrote:
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.

Godel used Principia as a basis for all such deductive methods.
That there are undecidable propositions within PM is clear.
The axioms of the system are examples.

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.

Principia mathematica, by Alfred North Whitehead ... and Bertrand Russell.

Twirlip;174158 wrote:

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!

I have not seen an appreciation of the foundations of mathematics compared to any sex life before, "ludicrous and unspeakably frustrating sex life" hahaha.
I'm happy to hear that you won't go into details, whewww! that was a close one.

---------- Post added 06-07-2010 at 10:16 AM ----------

Reconstructo;174148 wrote:
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.

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

A must read for foundational concerns, imo.
Introduction to Mathematical Philosophy
See his definition of natural numbers...

---------- Post added 06-07-2010 at 05:19 AM ----------

Reconstructo;174148 wrote:

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.

Epsilon is the base of the natural logarithms.
Yes. Sinh, cosh, and tanh, are the hyperbolic functions.

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?

---------- Post added 06-07-2010 at 05:22 AM ----------

Reconstructo;174148 wrote:

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.

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.

Twirlip

Mon 7 Jun, 2010 09:17 am
@Owen phil,
Owen;174185 wrote:
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.

I was referring, of course, to incompleteness, not inconsistency. I'm sorry if that wasn't obvious. It was only a passing remark in what was already a fairly jocular response to something that to begin with wasn't dead centre of the thread topic; so it's hardly worth fussing about.

Reconstructo

Mon 7 Jun, 2010 01:48 pm
@TuringEquivalent,
TuringEquivalent;174174 wrote:
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.

Oh, but there is an absolute library and more of mathematical opinion. And unless one loves the subject, why would one engage it in the first place?

Please, sir, if you have ideas to share, I would enjoy hearing them. This comment of yours is not to the point. I have read what I have read, about a thousand pages in the last few months. But there are millions more. And I am not bound to pursue this as you or anyone else might. My interest in as philosophical as it is mathematical. I am concerned especially with the intuition of unity. Unless someone addresses that, they are not yet TCB to my satisfaction.

---------- Post added 06-07-2010 at 03:01 PM ----------

Owen;174185 wrote:

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?

I don't know about K and religion. He was a great business mind who chose to do math rather than make an even bigger fortune. I think by the God line, he was saying that the integers were given to intuition, or something like that. As you prob now, he fought Cantor tooth and nail. He also loved to argue with Weierstrass.
I think K was something of a finitist and a constructivist. And computers cannot deal with the infinite and work by means algorithms -- which are constructive, I think.

---------- Post added 06-07-2010 at 03:04 PM ----------

Owen;174185 wrote:

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?

Ah, I do have a little experience with computers. Yes, and it's easy in a language to mix logic and mathematics. (If ((x * 3) > 34) Or Gameover then Gameend. (As you may know, Gameover would be a boolean variable and Gameend a procedure. Pascal was my favorite language, but I start like so many with BASIC.)

What you mention might already be invented. Do you know of Prolog?

---------- Post added 06-07-2010 at 03:07 PM ----------

Twirlip;174158 wrote:
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.

The first volume is at my library. I've sort of dodged it because other books seemed more interesting. I loved Chaitin's Metamath. AIT is great. I'm definitely more obsessed with the "absolute concept" as far as foundations go than any of the justifications for this or that. I want the root.

---------- Post added 06-07-2010 at 03:08 PM ----------

Owen;174185 wrote:

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.

Oh yes, I understand this completely. But it's one set of uncountable reals being greater than another set of uncountable reals that seems questionable. Doesn't this involve power sets? Every next aleph is a power set of the one before? Or something of that?

---------- Post added 06-07-2010 at 03:12 PM ----------

Owen;174185 wrote:

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

Well, for me it's about the number 1 or any particular synonym of unity, but rather that unity is something primary to human thought. Can we talk of unity or singularity except with synonyms? I did look at that link. Thanks.

---------- Post added 06-07-2010 at 03:15 PM ----------

Twirlip;174158 wrote:

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.

I suppose I am trying to point to something prior both to math and logic, founding both of them. It's something so essential and intuitive that I feel it is sort of glossed over. Number is essentially singular. Even 100 is one quantity, composed of 100 sub-quantities. And complex logical statements are unified by operators and brackets. We deal in unities. And this unity just is. As far as I can tell. It seems for any systems of symbols to be meaningful for us, we have to see them as individual singular symbols with one clear meaning.

---------- Post added 06-07-2010 at 03:17 PM ----------

Twirlip;174158 wrote:

Not at all, no.

Really? It seems to me that once one has the unit and positional notation, there is just equal steps of one as far up as one likes. For me, the naturals are just as convincing as logic. Logic is in its way perhaps more primary, and maybe more beautiful. But since logic can't offer quantity, math has a certain edge as well.

---------- Post added 06-07-2010 at 03:19 PM ----------

Twirlip;174158 wrote:

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!

Interesting! For me, the attraction of integers is in their ideal perfect clarity. The beauty of Chess is similar. A closed system that is ideally precise. Digital. The intuition is not bound to the paper. The paper is just a medium, or an aid to memory. Just as Chess does not require a board. But a board helps visualization.

---------- Post added 06-07-2010 at 03:21 PM ----------

The aesthetic effect on my is all positive. I wish this for all. But perhaps it makes me focus in less practical ways on the subject. For me it is more a body of sculpture than a tool. No one can cut stone as perfect as the number 1, or anyother natural number. It's meaning is ideal. Just as our spatial intuition can give us a perfect circle, so does intuitive/ideal arithmetic give us perfect ideal quantity. The digital/continuous divide. Both perfectly intuitive (for me) and perfectly beautiful, but they clash! Zeno is one of the reasons I got into math.

ACB

Mon 7 Jun, 2010 04:41 pm
@Owen phil,
Owen;174185 wrote:
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.

Reconstructo

Mon 7 Jun, 2010 04:57 pm
@ACB,
ACB;174387 wrote:
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.

Thanks for joining the thread. I too have my doubts about actual infinities. I also don't think we can really think them. And this ties into Kronecker and computing. And also the clash between the continuous and the discrete. There is a significant difference between a finite and an infinite set. Of course I enjoy the "conjuring trick" and find Cantor quite fascinating. But I see why K and others were skeptical. For me, the finiteness/precision of number is much of its appeal. THen we have numbers like e and pi that melt at the edges, but they are precise enough for practical purposes.

Alan Masterman

Sat 26 Feb, 2011 12:06 am
@Owen phil,
"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."

I'm not sure I could feel confident about that, Owen Phil, though I lack the mathematical knowledge to argue a case in detail... but I would draw an analogy with geometry. Euclidean geometry tells us that the internal angles of a triangle always add up to two right angles. But if we negate the "axiom of parallels", thus taking the turnoff into non-euclidean geometry, then it is no longer true that those angles will always add up to two right angles. Until about 1830, this was thought incredible; today we take it for granted.

In the same way, if we abolish the widely-accepted arithmetical axiom "0 is not the successor of any number" and replace it with "0 is the successor of unity", we would have an alternative arithmetic in which 1+1=0 (seemingly). As Russell might say, this would be very odd, and doubtless inconvenient, but it is not obvious that it would involve any purely logical absurdity.

1. Philosophy Forum
2. » Logic
3. » On the Foundation(s) of Mathematics