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 ----------
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 ----------
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 ----------
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 ----------
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 ----------
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 ----------
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 ----------
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 ----------
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.