I rather think that it's this way:
Gödel made just a statement and asked the machine whether his statement was true or not. It's like I would say the sky is green. And you would have to answer with true or false. No problem to answer to my statement with true or false.

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Beauty is the first test:
There is no permanent place in this world for ugly mathematics.

This reminds me of the book Gödel, Escher, Bach in which Douglas R. Hofstadter describes the following situation.

Whenever we try to model a system that will explain all possibilities of truth we must fail, and he uses the following example.

We will categorize all words into two categories:

A. words that are self-descriptive
B. words that are non-self-descriptive

Quintasyllabic . . . (group A)
word . . . (group A)
monosyllabic . . . (group B)
blue . . . (group B)
non-self-descriptive (group ?)

It won't fit into either category, whenever you try to put it into one category it wants to go into the other, I think it's pretty neat.

Gödel proved that whenever we try to come up with a system that will encompass all truths (as was thought to be probable by Bertrand Russell et al in principia mathematica) we must fail. Because it is possible to turn the system back onto itself (much as in the previous example).

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Math Guru

I disagree right back at you. I think that it depends on what you are measuring. For example, the concept of a fetus dividing into twins is NOT represented, in my perspective (used for agruement here only: I will attempt to show that math is totally objective), by 1+1=?. We have shown that in order to fertilize an egg, two reproductive cells (one being male and the other female, of course) must combine. I do not think of this as 1 cell + 1 cell because I am concerned with what happens to the embrio-to-be. I think of this as (chemicals in male cel) + (chemicals in female cell) = 1 embrio. The embrio's mass and chemical make-up is equivilant to the sum of those properties in the two reproductive cells: addition still makes sense looking at things my way. The situation with the worm can be approached in the same manner: (total mass of big worm) / 2 = (mass of new, smaller worm). (These aren't the best arguements, admittedly, but I think they get my point across.)

On the other hand, the two trees problem is different. You are trying to look at two different things at once. You are trying to say that 1(mass of tree 1) + 1(mass of tree 2) = 2(mass of either tree). In effect, you are saying x + y = 2x OR x+ y = 2y, niether of which makes sense mathematically since you cannot combine the two different variables.

In fact, I think that if you know how to set up the correct view mathematically, you can make sense out of any situation, such as the planting a tree problem.

Given: 1. A tree can create seeds capable of creating new trees from the new trees' immature form. 2. Trees grow by taking materials out of the ground and sucking up any water they can. 3. The youngest form of a tree is the same seed as a seed.
Prove: Trees do not somehow mysterously generate new trees in such a way as to force humans to create illogical mathematical expressions to decribe them. (This isn't an insult; Math Guru addmitted that this was what he / she was doing and used that as an arguement. This is my counter-arguement, as have been all of these examples.)

If I plant a tree, it will increase in size by collecting natural materials and building itself with them (given 2). Since the tree creates seeds (given 1), I can plant these and expect them to grow into new trees without increasing in size from the seed when planted (given 3). These new trees will build themselves by taking material from the ground (given 1). Therefore, trees do NOT create new matter or trees inexplicably, and can easilly be described mathematically and logically.

In my opinion, math is concrete. You just ned to know what you're looking at, and how to interprete it.

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"I am empowered to lie. But I have never lied to you." - The Giver

I think you got it right to say the Godel did away with the notion that mathematics was nothing more that a series of truths to be derived from a string of axioms accepted a priori. Suffice it to say that mathematics simply must be more than that, given Godel's theorem.

I would like to also add the following to the discussion. Last year I did a presentation on Godel's theorems. What I received was the objection from a number of people to the effect that "well, sure, you have this undecidable proposition... but its so artificially constructed and obscure that, who should really worry about such things?" In other words, from Godel's proof, one is left with the impression that incompleteness is a quite rare and pathological thing, and that most things in life are indeed decidable, and that Godel's theorem is anecdotal at best.

In the past few decades however, a new branch of mathematics called Algorithmic Information Theory arose, and seemed to be telling us quite the opposite. Instead, it seems that 'most' (under a suitable interpretation of the word most) propositions will be undecidable for a given set of axioms. With AIT, one is able to talk about the "information content" of a string of axioms, along with the information content of a given proposition. It follows readily that such a formal system cannot hope to prove propositions beyond a certain complexity. In the words of Chaitin, "10 pounds of axioms cannot prove a 20 pound theorem". Thus it seems that undecidability is a far more pervasive and systematic component of mathematics than we ever thought.

As for my personal opinion, I'm certainly not a formalist, but neither am I purely platonist either. On the one hand, I do not believe mathematics is purely a construct of our own imagination and being.... who could really be so conceited as to claim that he invented an equation as beautiful as e^i*pi +1 = 0? So I do believe in a mathematical reality outside of ourselves. On the other hand, I cannot believe either that mathematics is a reality completely divorced from human perception. If the universe found itself to be devoid of life, and the equation e^i*pi + 1 = 0 was left written on a blackboard with no one to comprehend or appreciate it, could one justifiably claim that mathematics any longer existed?

I guess what this amounts to saying is that I believe mathematics 'came' from somewhere/something/someone outside of ourselves -- but that it only lives in us.

I hope that wasn't too terribly convolluted. Write back if you have any thoughts.

Just a quick note,

I think that it is quite possible for man to create beauty. Mathematics seems to unfold as we discover it and as new branches of mathematics attract new questions and new discoveries. However beautiful and pure it seems, I beleive it is a product of man's logic. Maybe mathematics mirrors the construct of our own brain, and the beauty we see in mathematics a just a reflection of our own beauty.

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Math Guru

I don't think we 're that beautiful and I hope someone would vote "Formalistic" so that the poll doesn't end in a draw!

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“Mmmm, Beer” - Homer Simpson

If you are involved with survey statistics for any length of time you
will learn to judge what people think by their actions not by their
words.

Do Mathematicians behave as though they are playing a formal game
involving symbol maipulation, devoid of other meaning?

RonL

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Truth does not change because it is, or is not, believed by a majority of the people.

Giordano Bruno

Well... People have been known to overreact over many things devoid of meaning!

So I suppose the question is "can man perceive any glimpse of universal truth inside his own creations?"

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“Mmmm, Beer” - Homer Simpson

Now you are going to start PH off about football (sorry soccer) again

I think Mathematical Platonist would argue over "his own creations"
here, would they not? (not only over the gender specificity of the pro-noun )

Do they not regard mathematics as discovered rather than invented?

RonL

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Truth does not change because it is, or is not, believed by a majority of the people.

Giordano Bruno

He's a platonist, he's got every reason in the world to start arguing even without knowing why!

Absolutely.

And that's why Godel's theorems caused such an uproar. All mathematicians were (and still are) platonists in their very own way.

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“Mmmm, Beer” - Homer Simpson

Who am I, platonist or formalist?

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"Democracy has proved only that the best way to gain power
over people is to assure the people that they are ruling
themselves. Once they believe that, they make wonderfully
submissive slaves." - Joseph Sobran

If we are voting I cast mine in favour of the opinion that you are a formalist

RonL

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Truth does not change because it is, or is not, believed by a majority of the people.

Giordano Bruno

What about you CaptainBlank?

Tell me if I am wrong (I do not understand the exact meaning of formalist or plutonist).

Mathematicians are formalistic
Physicists (Applied Mathematicians) are plutonistic.

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"Democracy has proved only that the best way to gain power
over people is to assure the people that they are ruling
themselves. Once they believe that, they make wonderfully
submissive slaves." - Joseph Sobran

A plutonist - someone who believes that money exists in the world of "ideal; forms"? - No not I

A mathematical Platonist? Yes - I believe that certain abstract "entities" exist
independently of the believer.

RonL

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Truth does not change because it is, or is not, believed by a majority of the people.

Giordano Bruno

Hello, everybody here,

So finally we are talking about religion.

I don't want to go that far, so I consider Math as a kind of language, which is very convenient to describe and manipulate certain aspects of nature and culture (=sum of all human activities).
With this language you have signs (alphabets), a grammar and meanings. The language is an open system which can be enlarged easily if there is any need for it. Every branch of Math contains a nucleus with a problem that had to be solved.
If you don't know the signs or the grammar or the meanings then you can't detect the beauty of Math and you don't understand the results.

As with every language Math will die out if there aren't any "speakers" left who know to "speak" Math.

By the way: The same point of view you can use with painting, poetry, music and even social behaviour.

Last remark: I'm not a philosopher, for me it's sufficient that I can get results by using my very limited knowledge of Math.

Greetings

EB