As I've been tutoring in math lately, I've been thinking about what goes on within mathematics and possible relations to our own knowledge. Maybe some analogies here will clear up some of my ideas concerning the nature of logic and its relation to reality; on the other hand, maybe it'll just lead to more obfuscation.
(1) When we work with sines and cosines (as well as tangents, cotangents, secants, and cosecants), we are referring back to the relations between the sides of a triangle. We are actually talking about the sides of that triangle, with precise and true statements. However, we can never get at those sides in and of themselves. We really only get at their relations, and so all of our thinking about them is entirely relational. But this doesn't affect its validity. Further, through these trigonometric relations, we see all sorts of things which we would not have seen without looking at the relations; we can see how triangles relate to circles, we can work with all sorts of signals in engineering, we can calculate many different identities relating our relations are so delve into the depths of the nature of mathematics. All of this is possible precisely because we are working with the relations (we are speaking "analogically," if you will). So, even if we cannot know the sides in themselves, we can understand their significance and have real, true, precise, meaningful speech about them.
(2) We've all learned about the quadratic equation. There are cubic and quartic equations too, for higher order polynomials, and so anything with at most an x4 term in it can be solved through an equation (even if the equations are too complicated for most practical use). However, there is no general equation for any higher order polynomial; once an x5 is in there, kiss straightforward procedures goodbye. We can prove that there cannot be a general procedure for solving these monsters. I'll maybe look at the proof some other time; meanwhile, the importance of this is that even though we have a perfectly logical, ordered reality, we cannot analyze it all according to any one method.
Hurray for Platonism!
3 comments:
Nicely stated and illustrated!
I am ignorant in matters of mathematics and philosophy of mathematics. But I have this weak pre-theoretic notion that *all* the properties about a triangle would be reducible to relational properties about its sides. The relational properties are essential to the triangle, and each side considered independently and not as part of the triangle (not considered in relation to the other two sides) would not be the same entity, so it doesn't make sense to talk about the essential properties of the sides of the triangle apart from the relational properties of those sides.
I'm probably wrong for some reason; if you feel like instructing me, go right ahead.
Hmmm, I hadn't thought of that aspect of it before, SC. I think that you're on to something; I'll have to ponder it for a bit.
So, I propose the following change to the post: there are three sides, which happen to be arranged (right-)triangle-wise. We can talk about these sides given only their relational properties (such as being-in-a-triangle-arrangement and sines, cosines, tangents, &c.), and have meaningful speech, even if we were never able to talk about the sides directly. In fact, we would have meaningful, precise, propositionally true language about the sides through a contingent fact about them.
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