[quote]stokedporcupine wrote:
in an empty sense, you might say that all math deduces to addition (subtraction being addition of negative numbers). This though isn’t really correct, because you can’t explain algebra without a basic notion such as “equals added to equals are equals”–ie, without some more advanced set theoretic properties of equivalence relations (transitivity, reflexivity, and symmetry of equality). Further, algebra requires the notion of a variable, or a type of meta-symbol in mathematics. so really, everything after arithmetic requires these more advanced set-theoretic concepts. calculus is a whole different game, because somehow you must define the operations of integration and differentiation. this is pretty complicated once you get past the hand-waving done in normal calc classes about limits and reamen sums. i don’t really understand it all myself. Finally, geometry is a whole different ball game. I don’t know how much pure geometry you’ve done, but, systems of geometry can be constructed without any numbers at all. the definitions and proofs will look a little weird to someone accustomed to seeing metrical geometry, but they nevertheless work. so in this case, not only can geometry not be reduced to addition, it has no numbers at all.
anyway, all of this leads into a good point on the foundations of mathematics. That is, (1) numbers themselves cannot be defined purely in terms of addition, and (2) the set-theoretic functions that are the basis of algebra cannot be reduced to an addition function, and (3) axiomatics play a key role in the development of higher math (such as calculus and geometry).
Really, i suppose answering the question about whether or not math is about real things comes down to the question of whether or not numbers are real. But, defining just what a number is, is very difficult. the reason i mention all of the functions and axiomatics is that often these things themselves are used to try and define “number” contextually. i’m also trying to be fair here, and give you the stuff mathematicians care about when they discuss this stuff. (unlike what some people think, i really do try to be fair and complete when i talk about things)
pure philosophers, as you can imagine, generally only care about what “numbers” are, and whether numbers really exist. this is probably the part you really want to hear about anyway, so i’ll try to overview a little.
Numbers are generally considered to be abstract entities, or “universals”. that is, the number 2 is merely abstracted away from many particular instances of 2 (like seeing 2 sheep, hearing 2 sounds, seeing 2 colors, etc…). The question of whether universals really exist or not (or, in Plato’s terms, the question of whether Forms exist, though i hesitate to make anymore then a casual link here, because plato would reject the above account of number for many reasons) is huge. while you might deny that the universal form of “2”, the number itself, exists, and claim that only particular instances of 2 exist (and further that the universal 2 is merely a product of human imagination, just like unicorns), you run into some problems with such a claim. this is a complicated subject that i’m not really prepared to discuss in detail (i don’t want to say misleading things).
the general problem though is that rejecting the universal form requires showing how the particulars can perform the same theoretical functions as the universals. in simpler terms, rejecting the existence of universal numbers requires explaining mathematics in terms of only particular instances of the numbers.
anyway, i’m sorry for not having more time, i love math. ha. anyway… given your background you should be able to handle any of the stuff i suggested.
also, i forgot about this site when i typed my first response. you might find these two articles interesting:
http://plato.stanford.edu/...sm-metaphysics/
http://plato.stanford.edu/...hy-mathematics/
hope you find those helpful.[/quote]
Haven’t forgotten about this one, just trying to read a bit before responding.
You’re definitely correct that math can’t be reduced to just addition. Whether this presents a real hurdle to it just being imaginary I’m still unclear on. I read the links and I seem to agree with Intuitionism, minus the ideal mathematician… seems extraneous so far, but I’m still reading. As to the difficulties this view presents, the article was very vague, only saying that they exist at the higher levels of math. Over the next few weeks I’ll read up on what exactly those difficulties are (this stuff takes a long time to digest). My hunch now, is that these difficulties only arise when math is used to model something that can’t exist… like an 11 dimensional object for lack of a better example.
If that turns out to be the case, I think it would be a more of a knock against the way math is practiced than against the view that numbers are imaginary. It would be like reformulating algebra because it couldn’t tell you how many angels can dance on the head of pin.
Just my thoughts for now.