311–320 of 322 posts
The question assumes that there is a difference between invention and discovery. There isn't. Invention is a kind of discovery. The apparent distinction arises from our separation of physical and mental processes. But this is a purely artificial separation, a human conceit, because (unless you're a dualist) mental processes are physical processes. Specifically, mental processes are computational processes, which are…
I am not sure of the implications. Does it even matter if Math is invented or discovered? Maybe it's both, and it isn't a contradiction between invention and discovering? We describe things having a certain radiation wavelength as having the color yellow. In that sense, yellow it's an invention. That doesn't mean the radiation doesn't exist. But to complicate things a bit, some things don't exist unless we observe th…
I figure reality is part of math so if it was purely invented we wouldn't exist.
There's one thing I cannot wrap my head around. Math is purely abstract science yet it describes the world around us to the utmost precision. Does that mean that this world is simply ... a math model which means everything around us is ... not real? I've long stopped believing in free will because everything points at it being an illusion of our brain because we're a product of this world and we had no chance of infl…
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And yet, most blind mathematicians are geometers, which suggests your theory is incorrect.
I'm not convinced that blind mathematicians count as their own species, nor that they echolocate, and so it doesn't seem to say much about his theory one way or the other.
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This. It is discovered in the same sense that humans "discover" facts about humans via science- we're not born with perfect self-knowledge. It is "created" because there is no guarantee that our efforts correspond to reality. In that sense we are just playing in the sandbox of what we can conceive. The fact that mathematics corresponds so "unreasonably" to objective reality is because what we call objective reality i…
>It is "created" because there is no guarantee that our efforts correspond to reality. In that sense we are just playing in the sandbox of what we can conceive. But doesn't this dance kinda near the notion that I am a brain in a vat and none of you exist (read this question with me as the speaker or with yourself as the speaker)? I would say our brains do have limits. I can't well envision a 4D object. But once we re…
I agree.
But this assumes logic and at least the principle of non-contradiction: not both A and not A - is how reality "really" is.
We can't get past the idea that it must be this way because only provable nonsense lies on the other side of this assumption. But our brains may be fundamentally unable to process ultimate reality and the nonsense a contradiction represents may be a statement not about reality but our brains, our thinking.
So logical contradictions aren't actually nonsense, they're the sound of us hitting the walls of what our minds can conceive of. All animal have such limits. We assume those limits look like darkness- stuff we can't peer into. What if they look like impossibility instead?
That's the point of view I'm entertaining here. I am not saying this is true. It certainly isn't useful or provable as far as I know, but it is possible.
The practical value of such an exercise, if it has any (and I think it does) is to twofold.
One seems to expand my imagination to the maximum extent pops me out of the assumptions that frame my thinking and this seeps into my thinking about things, technical problems, generally.
Two it confers humility and a certain openess and makes me less judgmental. So that, for example, when I hear or read people with claims to spiritual knowledge I don't automatically blow them off as crazy / bitter / ignorant because what they're saying "makes no sense".
Thinking and talking about Ultimate Reality capital U capital R, ought to fill us all with humility if we're being intellectually honest by our own standards. Yet, I find people totally lack that humility. They make huge pronouncements about Ultimate Reality which they can't really be sure of, and the effect this has on the world, and how we think of each other, and therefore how we treat each other, and even the effect on one's own mind, is one of diminishment generally.
Mea culpa, I was one of those people and I didn't like it.
If you get down to the core of knowledge and how we know something, this is what's really there, and it's good to be reminded of it.
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You shouldn't be mapping propositions, you should be mapping theories. See my comment below on the internal logic of a topos. What you are referring to as a proposition is an element of the Heyting algebra or the Boolean algebra.
Why do you think the same thing doesn't apply to theories? Since you mention topos theory, the double-negation topology is actually a basic construction in topos theory, used to construct a dense Boolean subtopos of a given topos.
My point is something else: I don't know of a way to have a intuitionistic exterior logic and a classical internal logic.
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All you did was discover that if you typed those words on your computer, they would appear on that website on that day.
So each distinct event can be discovered, regardless of how predictable it is based on previous events? Each sunrise is a discovery?
To a scientist/saint who has done a lifetime of study/penance on the past, present and future, every invention is predictable. To a goldfish with 5-second memory, every sunrise, why every breath, is a discovery.
Look at it another way. A scientist 'discovers' the atom. He didn't 'invent' it, right? Now, how did he discover it ? Using a microscope that another scientist 'invented'. So far so good?
Now go down the chain to that scientist. How did he 'invent' the microscope ? Did he make it out of thin air ? No, he was one day playing with 2 optic lenses that another scientist 'invented', and when placed in a line, he 'discovered' that they magnified it 100 times what one lens did. And he told everyone that 2 lenses together is called a microscope that he 'invented'. So far, so good ?
Now, go down the chain to that scientist. How did he 'invent' the optic lens ? Did he create it out of thin air ? No, he was one day playing with glass which another scientist 'invented', and when looking through it, he 'discovered' that they magnified it things 10 times their size. And he told everyone that looking through a glass lens is a 'magnifying glass' that he 'invented'. So far, so good ?
You see the pattern here ? Every permutation combination of nature exists. In other words, all possible inventions already exist waiting to be discovered by humans, or all possible discoveries are waiting to be invented.
The 2 words are synonyms.
Something 'new' that you 'discovered' was someone else's 'invention'. And once you go beyond the deepest chain that your senses can reach, even the physical laws that govern our universe that were 'discovered' are an 'invention' of God. Where did God discover it from ? We can ask Him.
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> It's silly to think that the sqaure root of negative one is something real to be discovered. As you said, the complex numbers are the unique (up to isomorphism) algebraic closure of R. Given that they arise as the unique solution to (in my opinion) a pretty fundamental question about the real numbers, I think it's fair to say that they've been "discovered", not "invented".
Real numbers might be less real than you think. Once you get beyond the number of possible quantum states in the universe, what do higher numbers really mean?
2. Yes, real numbers, much like complex numbers, were "invented". But complex numbers lay hidden, waiting to be discovered, as the algebraic closure of the reals, and similarly, the reals can be discovered from the "simpler" ideas of ordered field and Dedekind-completeness.
3. That's the weird thing about mathematics --- when you invent things, you leave a world of discoveries for others to make, and sometimes those discoveries are that your invention has inside it a perfect mirror image of another invention.
4. Once you leave classical logic, you suddenly have a lot more room for invention, because there are several competing definitions of "real number", and none is definitively better.
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So each distinct event can be discovered, regardless of how predictable it is based on previous events? Each sunrise is a discovery?
"predictability" is only a function of your memory capacity and/or knowledge. To a scientist/saint who has done a lifetime of study/penance on the past, present and future, every invention is predictable. To a goldfish with 5-second memory, every sunrise, why every breath, is a discovery. Look at it another way. A scientist 'discovers' the atom. He didn't 'invent' it, right? Now, how did he discover it ? Using a micr…
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It's a common theme among sculptors too: the shape is already in the stone, they just have to uncover it.
Is it? The only time I have ever heard this expressed is as a bad joke.