\nThroughout elementary and high school, I got awful marks in math. I always
\nassumed I was just stupid in that way, which is perfectly possible. I also
\nhated my teacher, so that didn’t help. A friend of mine got his PhD in math
\nfrom Harvard before he was 25 (he is in his 40’s now) I was surprised the
\nother week when I learned he isn’t particularly good at basic arithmetic etc.
\nHe said that’s not really what math is about. So my question is really for
\nmath fans\/pros. What is math, really? I hear people throwing around phrases
\nlike “elegant” and “artistic” regarding math. I don’t understand how this can
\nbe. To me, math is add, subtract, etc. It is purely functional. Is there
\nsomething you can compare it to so that I can understand?\n<\/p><\/blockquote>\n
This hits on one of my personal pet peeves. Math really is a beautiful
\nthing, but the way that math is taught<\/em> turns it into something
\nmechanistic, difficult, and boring. The person who posted this question
\nis a typical example of a victim of lousy math education. <\/p>\n So what is math? It’s really a great question, and not particularly
\nan easy one to answer.<\/p>\n
<\/p>\n
You’ll get lots of different answers depending on just who you
\nask. It’s a big enough thing that you can describe it in a lot of
\ndifferent ways, depending on your perspective. I’m going to give
\nmy own, and you can pipe in with your own in the comments.<\/p>\n
To me, math is the study of how to create, manipulate, and understand
\nabstract structures. I’ll pick that apart a bit more to make it more
\ncomprehensible, but to me, abstract structures are the heart of it. Math
\ncan<\/em> work with numbers: the various different sets of numbers are
\nexamples of one<\/em> of the kinds of abstract structures that we can work
\nwith. But math is so much more than just<\/em> numbers. It’s numbers, and
\nsets, and categories, and topologies, and graphs, and much, much more.<\/p>\n What math does is give us a set of tools for describing virtually
\nanything<\/em> with structure to it. It does it through a process
\nof abstraction<\/em>. Abstraction is a way of taking something
\ncomplicated, focusing in on one or two aspects of it, and eliminating
\neverything else, so that we can really understand what those one
\nor two things really<\/em> mean.<\/p>\n For example, look at topology. Topology is basically a way of
\nunderstanding shapes. But it does it in a completely abstract way. It throws
\naway everything except the concept of closeness<\/em>. You have a
\ncollection of points, and you’ve got a concept of things that are
\nclose<\/em> to one another, defined in terms of neighborhoods<\/em>. By
\nplaying with different notions of what things are close to each other, you can
\ncreate any shape you can imagine, and some that you probably can’t. But you
\ndon’t really need numbers at all: you can just create and play with shapes in
\ntopology – as long as you’ve got the set of points, and you’ve got set
\nrelations, you can figure out what it really means for something to be a
\ntorus. You can see what’s really strange about a moebius strip. You can
\ntake the moebius strip, and add a dimension to it, and see exactly how you
\nproduce a klein bottle. <\/p>\n For another example, look at category theory. It’s a way of understanding
\nfunction. What’s a function? At it’s core it’s a mapping<\/em> from one
\nthing to another. But what does that really mean? What can you do<\/em>
\nwith that basic idea? What can you make<\/em> with it? The answer is:
\nvirtually anything you can imagine. <\/p>\n But math is more even than just those abstract things. Why does music
\nsound good to us? Because it’s got an underlying structure. That structure
\ncan be described mathematically. Personally, I’m a huge Bach fan. I believe
\nthat he was the greatest composer of music that ever lived. His music
\nis magnificently beautiful, and incredibly moving. But to really understand
\nit, to really grasp all of what he was doing in his music, you need to understand
\nthat it’s structure on structure on structure on structure. That structure
\nis mathematical. If you’re really understanding the structure of Bachs music –
\nif you sit down and analyze it, you’re doing math<\/em><\/p>\n When you look at a cubist painting, you’re looking at a strange kind of
\nprojection of something. The artist has taken the subject of the painting
\napart, viewed it from different perspectives, different points of views,
\ndifferent ways of understanding it or seeing it, and assembled them together
\ninto a single image. When you look at a cubist painting, and try to understand
\nwhat the artist was seeing, how they were seeing it, and how the pieces
\nof the final image really fit together – you’re doing math<\/em>.<\/p>\n When a scientist tries to analyze something about the world, to understand
\nhow it works, and describe it in a way that tells us something important about
\nhow things behave – they’re doing math. They’re abstracting<\/em> the
\nworld to come up with a precise, formal, descriptive way of stating what
\nthey’ve learned. <\/p>\n When you look at a road map, and figure out how to get from one place to
\nanother – you’re doing math. The map is an abstract<\/em> representation of
\nthe world that allows you to do certain useful things with it. (And frankly,
\nthis is one example that I’ve never been able to understand. I can’t read
\nmaps. Quite literally a bit o’ brain damage – some scar tissue in the left
\nfrontal lobe of my brain.)<\/p>\n When a jazz musician improvises, part of what they’re doing is
\nmath. For an improvisation to make sense, for it to sound good, and fit
\nwith what’s going on around it, there are a set of constraints on it:
\non pitches, pitch progressions, rhythm, chords. Those are all abstract
\nproperties of the music, which are mathematical!<\/p>\n
Math is unavoidable. It’s a deeply fundamental thing. Without math,
\nthere would be no science, no music, no art. Math is part of all of those
\nthings. If it’s got structure, then there’s an aspect of it that’s
\nmathematical.<\/p>\n","protected":false},"excerpt":{"rendered":"
I’ve got a bunch of stuff queued up to be posted over the next couple of days. It’s been the sort of week where I’ve gotten lots of interesting links from readers, but I haven’t had time to finish anything! I thought I’d start off with something short but positive. A reader sent me a […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"jetpack_post_was_ever_published":false,"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"_jetpack_memberships_contains_paid_content":false,"footnotes":"","jetpack_publicize_message":"","jetpack_publicize_feature_enabled":true,"jetpack_social_post_already_shared":false,"jetpack_social_options":{"image_generator_settings":{"template":"highway","default_image_id":0,"font":"","enabled":false},"version":2}},"categories":[74,24],"tags":[],"class_list":["post-832","post","type-post","status-publish","format-standard","hentry","category-basics","category-goodmath"],"jetpack_publicize_connections":[],"jetpack_featured_media_url":"","jetpack_shortlink":"https:\/\/wp.me\/p4lzZS-dq","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/posts\/832","targetHints":{"allow":["GET"]}}],"collection":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/comments?post=832"}],"version-history":[{"count":0,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/posts\/832\/revisions"}],"wp:attachment":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/media?parent=832"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/categories?post=832"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/tags?post=832"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
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<\/p>\n