[quote]mbm693 wrote:
stokedporcupine wrote:
Again, this is a view which needs to be substantiated. also, your view is not the standard view in math or philosophy–actually, all the “evidence” seems to suggest otherwise, that math is not merely a human construct. i don’t really have time to go into all the details, but, suffice to say you would have a very hard time defending this point.
I feel pretty good about this one. I’ve googled and I haven’t been able to find any discussion of whether number are real or not (Plato’s forms was as close as I got). Do you have some resources you can point me towards or a more specified argument?
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This isn’t really something you “google”–there isn’t really much by way of non-technical discussion on philosophy of math or the foundations of arithmetic. (whichever you’d like to call it) You are right though in assuming that Plato’s theory of forms is somewhat connected to the subject. Plato and Aristotle actually have fairly sophisticated mathematical views, although they are generally not easily accessible. It takes quite a bit of digging and cross referencing to piece together their views on mathematics. (this is also because the general view of numbers held by the Greeks was different then our own.)
As far as sources, I can give you both ancient and modern stuff. On the ancient stuff, you can search amazon for books on Plato’s “late ontology”. Mathematics generally play a key role on what’s considered Plato’s mature or late theory of forms. If you have access to philosophy journals, do a search for the same thing. If you do the amazon search, i recommend you look for stuff by Sayre, he’s quite good. (generally, the people with backgrounds in math and logic are a better source for ancient number theory then people with backgrounds in classics).
The ancient stuff, though interesting, is fairly impoverished compared to the modern stuff. Modern interest in the foundation of arithmetic arguably started with Frege. He’s book “The foundations of Arithmetic” is a classic on the subject (though his thesis is mostly rejected by modern scholars). Another good source, which is somewhat more accesssable then Frege, is Bertrand Russell. His book “the principles of mathematics” is another classic on the subject, though again still a bit outdated.
Contemporary authors on the subject include those like Kit Fine and Stewart Shapiro (amazon searches should turn up quite a few books by them, some very technical, others not). There are many others, but, these are some of the bigger names in the field.
Like i said though, the only problem is that most of the material on the subject assumes the reader has a detailed background in formal logic, set theory, boolean algebra, etc… just read the descriptions carefully, you can find non-technical books.
I’m not sure about wikipedia, i never looked at any of the pages on the subject.
hope that helps.
Edit:
I won’t try to summarize anything here, as i don’t know your background at all. But, I’ll say this.
Mathematicians themselves generally say that they are not “inventing” anything per say, but rather discovering things. The notion that mathematical knowledge is objectively true, and universal, is well accepted. this is born out even in every day experience–the mathematical systems developed by different cultures have always been compatible–ie, when the Indians added 2 and 2, they got 4 too, just like the Greeks and the Egyptians. this suggests that mathematics is not mere invention, but something much more.
there is of course always the debate over whether mathematics can be reduced to pure logic, or whether it is its own separate field. those like Russell and Frege tried to prove the former. David Hilbert himself at first thought the same thing. After Godel’s work, most people reject that mathematics were completely reducable to logic.
Basically, there are two central questions to the philosophy of mathematics: What are the objects of study in mathematics? and what is the nature of mathematical operation. Answering your question of whether mathematics is “real” involves answering these two questions.