peoplater

So we all agree then. I am right and the only philosophy of math that makes any sense is the one that says mathematical concepts are just generalizations of everyday physical intuition. Everyone who says otherwise just doesn't know any better.

right. what did I say?

I looked at the construction and it is very similar to the way surreal numbers are constructed, but we are still comparing the same type of thing which is an ordered pair <x,y>.

They both make valid points, one is a constructive mathematician and the other is in some sense a formalist, and both of their areas of research are now flourishing mathematical areas. And yes they are both right.

But that is my point. Their writings, the ones I know of anyway, are so mathematically flavored and use so much mathematical intuition that they can hardly be considered philosophy. Any math student after spending some time with certain mathematical concepts would come to the same conclusions.

Also, if anyone is interested, I found yet another construction of the reals from just the integers. Here is the website http://arxiv.org/PS_cache/math/pdf/0405/0405454.pdf

If you don't mind, could you provide me with the ordering relation on such a set. The set would be composed of two different types of entities, and I do not see how you could define an ordering relation on such a set.

I still stand by what I said before. "Nature" is a very well known for publishing results that are in some sense groundbreaking and if his theory is so groundbreaking that it is going to revolutionize physics and energy production then he should submit his work to "Nature". But he hasn't done that. If he has done what he claims to have done then soon enough his work will attract the attention of well established physicist and we'll go from there, but I would not bet on that one happening.

Great. You made my point quite well. But I don't understand what you mean by the above paragraph.

You don't understand what my point is and keep pointing out that none of my assertions have any basis. In fact the basis is my personal experience and the books I have read on the matter. So here is my point. To talk about any subject, one should at least understand it and have some kind of intimate knowledge of the subject. Such knowledge comes after years and years of being immersed in the subject. Most philosophers do not have such a background in mathematics, and that is why when they "opine" on the matter, their "deep" philosophical inquiries seem quite childish. Plus, all the great mathematicians that were also philosophers, had very little to say on the subject of philosophical foundations of mathematics. If you read any of their works you will notice this. Most of their writings deal with logical foundations and there is very little actual philosophical inquiry. Most of their writing deals with trying to provide a foundation that is free of paradoxes. The little that does deal with philosophical foundations of mathematics is simple and nothing special.

If you look at the construction of the the integers from the naturals, then the construction of the rationals, from the integers, and so on up to reals, you will see that it is not the same equality. p/q=m/n whenever pn=qm, equality in the rationals involves more than equality for the integers, one is defined in terms of the other. Equality for the reals involves even more concepts than simple multiplication, like in the case of the rational numbers. One can define reals as sequences of rationals or as dedekind cuts, most books define the reals in terms of sequences of rationals and one says that two real numbers are "equal" whenever the sequences that define them have the same limiting value. So when one says .9999999999... = 1, if one interprets the equals sign as equality for integers, the the assertion is obviously false, but when one spells out the details and says that the real number .99999999... defined by a certain sequence and the number 1 defined by another sequence have the same limiting value then there is nothing mysterious about the assertion .9999999... = 1. The whole thing relies on people not knowing enough about sequences and series to interpret the equality sign in the right way.

As Hal mentioned, Russel, Kronecker, Hilbert were all great mathematicians but they were not such great philosophers, and many of them switched their stances on mathematics throughout their years. Also, there are many great philosophers, the greatest in my opinion being Wittgenstein, who did not know enough math to be able to talk about it in any non-trivial manner, but that did not stop them from doing so. My point in all of this is that for some reason or another non-mathematicians think that they know enough to provide a philosophical foundation for mathematics. That is why I say one should at least know enough algebra and topology to be able to understand non-trivial results in algebra and topology, once you know that much then go ahead and say math is this or that. Besides the thread was about an alternative to platonism in mathematics and my stance of mathematical ideas and concepts being some generalization of physical intuition, I think provides a great alternative, because I do not posit an existence of some realm that no one can touch or feel.

It is relevant because high energy physics is where new particles are discovered. And for someone to create a new particle, then they have to be doing some kind of high energy physics, because ripping electrons and protons is not an easy task. So if there was such a particle as a "hydrino" then the people working with the supercolliders would have noticed it before him. Second, most theoretical physicists do predict the existence of new particles and other such things, but most of their results are based on sound theories. This guy says he has discovered a new quantum theory without showing how his results agree with experimental things that are happening right now. That is why his new "theory" sounds suspect to me. Yes, someone has to be the first to discover something new, but it takes a great deal of training to be able to talk about physical theories, especially quantum physics and particle physics, so for all those other people who certainly have more training than this guy in physics to not have noticed something like that is highly unlikely. That is what I meant.

I very much agree with most of what Hal has said and on the issue of intuition and imagination in solving mathematical problems there is a great book by J. Hadamard titled "The Psychology of Invention in the Mathematical Field". The author discusses what happens in his mind when he is doing mathematics and compares his thought process to the thought processes of other mathematicians. Not surprisingly many mathematicians think in vague images, very few of them think in exact symbolic terms. Most of them even admit to just finding the solution of a problem just out of nowhere, after working in it for a very long time. That is why it seems to me to be pointless to subscribe to any kind of philosophy of mathematics and ask questions like "what is a number?", "where does it come from?" and so on. All mathematical things are just a byproduct of the way our minds are put together. The idea of a number is inherent in the construction of the brain. I do not think it can be described in any other terms.

I looked at the first chapter and it does not appear any simpler than regular trigonometry. It replaces 2 undefined terms for two other undefined terms. I do not see how this leads to any simplifications. The amount of work involved in getting answers from data is still the same, and in one example he got two answers and decreed that the first one was the obvious choice. His explanation did not hint at why the first one was the obvious choice. If anyone here buys it and actually goes through with it and learns the material then please post a review for the rest of us. As it stands I would not buy this book.

You seem like a smart guy and like many other smart people you have the same problem. For a second, just forget about everything, and by that I mean money, fame, power, recognition, failure, success and all the other stuff that goes with being smart. Once you do that ask yourself this question, "What is my passion?". If you can answer that question then you will be set.

Just another crackpot pretending to have invented another renewable energy source and that everyone in the established physics community is stupid for not looking at his results, blah blah blah. If it was possible to do what he says he can do then other and much more brilliant physicists would have figured it out before him, especially since most of them do high energy physics with supercolliders day in and day out.

Dudes and dudettes. The problem is not differentiating between '=' for real numbers and '=' for integers. Equality for real numbers is different from equality for integers. The proof relies on the fact that people interpret '=' in the sense of integers and in the middle the author of the proof changes his interpretation to '=' for real numbers. The proof relies on the fact that '=' is an overloaded relation.

It doesn't differ in any way from Objectivist, at least I don't think it does. By intuition I mean the images that are conjured up when you see a+b=c. Even the greatest of mathematicians do not think in totaly abstract terms. There is some image in their minds when they are trying to prove or disprove something, and that image does not just come out of nowhere. It comes from their years of experience in life and in mathematics. One more point. The problem is that it is philosophers that do philosophy of mathematics and not mathematicians. To speak about mathematics one should at least know enough to understand some non-trivial results in algebra and topology, but I do not think there are any such philosophers, and most mathematicians do not need philosophical justification to do what they do.

Everyone, everyone, can I have your attention please. All philosophers of mathematics and science in general can stop asking any more questions because I have figured it out. It is impossible to do math or physics or chemistry or anything else without having it in some way grounded in reality. So it does not matter if you are an objectivisit or a platonist or anything else, if you are doing math then your intuition is necessarily grounded in reality.