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Pure math concepts like complex numbers are not naturally existing, just waiting for us to find them, they're human-defined tools to describe things. Like new

by count 5y ago

Pure math concepts like complex numbers are not naturally existing, just waiting for us to find them, they're human-defined tools to describe things. Like new words, they're invented. They wouldn't be there without us, as they are, for the most part, artifacts of our cognition.

pohl 5y ago
If that were the case, would you expect another civilization on some rock in different galaxy to arrive at entirely different concepts, or ones isomorphic to our own?
- delecti 5y ago
  That's a pretty hard question to answer, and I think impossible to answer definitively (at least unless we met a civilization that did arrive at different concepts). It's kinda like the allegory of the cave; it's hard to envision another way of looking at the world, but that doesn't necessarily mean it's impossible for there to be one.
  - JackFr 5y ago
    It’s a good question to think about. A constant refrain of mine is that our brains have convinced us that they’re universal understanders, but we really don’t know that to be true. Imagine the difficulty of dealing with aliens who have a different mapping of the physical universe, different than our mathematics, which is both true but literally and physically incomprehensible to us.
- robotsteve2 5y ago
  Their observations of nature and their ability to predict stuff should be consistent with ours. They might not use the same mathematical tools or the same physical models, but they should make the same predictions. That is, we might not be able to understand their theory of gravity, but whatever their theory is, it has to be able to make predictions about orbits, black holes, etc. I don't think we could assume much about the mathematical concepts they use beyond that.
- anchpop 5y ago
  I’d expect other civilizations to have also invented rockets, microwave ovens, radio communications, and more. Does another civilization arriving at the same thing as us mean it isn’t an invention?
- ncmncm 5y ago
  Their mathematics will most likely have fundamental differences from ours that have led them to solve problems we haven't noticed. They will be appalled at how we have missed such basic features of the universe. And we will be appalled at some things they haven't noticed.
  - deleted 5y ago
    [deleted]
jonbronson 5y ago
That's self-evidently untrue. The properties that make a circle would be true regardless of whether a human ever set eyes on a perfect circle. Us identifying those properties is an act of discovery via research. Codifying those mathematical truths into a written notation is the only component of the process that could really be called invention.
- UncleMeat 5y ago
  But we still chose the labels. We chose what the elevate to the level of a mathematical object. Heck, even the idea of "what is true" is not universal in mathematics. Intuitionist and classical logic have different ideas of what. it means.
- b3kart 5y ago
  No human has ever set eyes on a perfect circle, because it (most likely) doesn't exist in nature. As such, I would argue that a perfect circle is a concept (or a model, if you will) that we've invented to make it easier for us to deal with an imperfect real world. I would not call identifying properties of such a model an act of discovery: one can come up with any model, no matter how far from reality, and use some set of axioms to identify its properties, but none of it will make said model real or fundamental in any way. The best we can hope for is that the model will be useful for making predictions about the real world.
- jhedwards 5y ago
  It's not self-evidently untrue, this is an incredibly complex philosophical question with no easy "right answer". There is also a spectrum of positions about this. The Stanford Encyclopedia of Philosophy is a good resource for reading about this topic: https://plato.stanford.edu/entries/platonism-mathematics/#ObjMatPla https://plato.stanford.edu/entries/platonism-mathematics/#Ob... Edit: another good link: https://plato.stanford.edu/entries/nominalism-mathematics/ https://plato.stanford.edu/entries/nominalism-mathematics/
- moffkalast 5y ago
  Well it really depends on how far you push the definition. There are inherent properties that can be discovered, but the way they're calculated and described is purely arbitrary. You need to get the same result of an area of a circle in the end, but how you get there is invented. Far more feasible and evident for complex stuff than the basics of course. As shown in the video, the depressed quadratic was basically solved by 3 people in 3 different ways, with today's description and definition being different from that too.

by count 5y ago

pohl 5y ago

If that were the case, would you expect another civilization on some rock in different galaxy to arrive at entirely different concepts, or ones isomorphic to our own?

delecti 5y ago
That's a pretty hard question to answer, and I think impossible to answer definitively (at least unless we met a civilization that did arrive at different concepts). It's kinda like the allegory of the cave; it's hard to envision another way of looking at the world, but that doesn't necessarily mean it's impossible for there to be one.
- JackFr 5y ago
  It’s a good question to think about. A constant refrain of mine is that our brains have convinced us that they’re universal understanders, but we really don’t know that to be true. Imagine the difficulty of dealing with aliens who have a different mapping of the physical universe, different than our mathematics, which is both true but literally and physically incomprehensible to us.
robotsteve2 5y ago
Their observations of nature and their ability to predict stuff should be consistent with ours. They might not use the same mathematical tools or the same physical models, but they should make the same predictions. That is, we might not be able to understand their theory of gravity, but whatever their theory is, it has to be able to make predictions about orbits, black holes, etc. I don't think we could assume much about the mathematical concepts they use beyond that.
anchpop 5y ago
I’d expect other civilizations to have also invented rockets, microwave ovens, radio communications, and more. Does another civilization arriving at the same thing as us mean it isn’t an invention?
ncmncm 5y ago
Their mathematics will most likely have fundamental differences from ours that have led them to solve problems we haven't noticed. They will be appalled at how we have missed such basic features of the universe. And we will be appalled at some things they haven't noticed.
- deleted 5y ago
  [deleted]

jonbronson 5y ago

That's self-evidently untrue. The properties that make a circle would be true regardless of whether a human ever set eyes on a perfect circle. Us identifying those properties is an act of discovery via research. Codifying those mathematical truths into a written notation is the only component of the process that could really be called invention.

UncleMeat 5y ago
But we still chose the labels. We chose what the elevate to the level of a mathematical object. Heck, even the idea of "what is true" is not universal in mathematics. Intuitionist and classical logic have different ideas of what. it means.
b3kart 5y ago
No human has ever set eyes on a perfect circle, because it (most likely) doesn't exist in nature. As such, I would argue that a perfect circle is a concept (or a model, if you will) that we've invented to make it easier for us to deal with an imperfect real world. I would not call identifying properties of such a model an act of discovery: one can come up with any model, no matter how far from reality, and use some set of axioms to identify its properties, but none of it will make said model real or fundamental in any way. The best we can hope for is that the model will be useful for making predictions about the real world.
jhedwards 5y ago
It's not self-evidently untrue, this is an incredibly complex philosophical question with no easy "right answer". There is also a spectrum of positions about this. The Stanford Encyclopedia of Philosophy is a good resource for reading about this topic: https://plato.stanford.edu/entries/platonism-mathematics/#ObjMatPla https://plato.stanford.edu/entries/platonism-mathematics/#Ob... Edit: another good link: https://plato.stanford.edu/entries/nominalism-mathematics/ https://plato.stanford.edu/entries/nominalism-mathematics/
moffkalast 5y ago
Well it really depends on how far you push the definition. There are inherent properties that can be discovered, but the way they're calculated and described is purely arbitrary. You need to get the same result of an area of a circle in the end, but how you get there is invented. Far more feasible and evident for complex stuff than the basics of course. As shown in the video, the depressed quadratic was basically solved by 3 people in 3 different ways, with today's description and definition being different from that too.