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Is Mathematics Discovered Or Invented?
Posted by
kdawson
on Sat Apr 26, 2008 05:42 PM
from the plato-says-yeah-but dept.
from the plato-says-yeah-but dept.
An anonymous reader points out an article up at Science News on a question that, remarkably, is still being debated after a few thousand years: is mathematics discovered, or is it invented? Those who answer "discovered" are the intellectual descendants of Plato; their number includes Roger Penrose. The article notes that one difficulty with the Platonic view: if mathematical ideas exist in some way independent of humans or minds, then human minds engaged in doing mathematics must somehow be able to connect with this non-physical state. The European Mathematical Society recently devoted space to the debate. One of the papers, Let Platonism die, can be found on page 24 of this PDF. The author believes that Platonism "has more in common with mystical religions than with modern science."
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Firehose:Is Mathematics Discovered or Invented? by Anonymous Coward
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Logical positivism to the rescue... (Score:5, Insightful)
When faced with an awkward question, logical positivism [wikipedia.org] asks: what would the answer tell me about the future?
Suppose you had a definitive, 100% guaranteed answer to the "discovered vs invented" question. What would it allow you to do that you couldn't do before? What could you predict? What would you gain?
Nothing, nothing and nothing.
It's meaningless; merely a matter of perception, wordplay and people having too much time on their hands.
Oh, and the correct answer is "discovered".
Re:Logical positivism to the rescue... (Score:5, Insightful)
No, the correct answer is "both."
The relationships and observations that we use mathematics to model are discovered. They are out there, we discover them, and then we model them. That should be obvious to all but the most die-hard of idealists.
The language that we use to do this modeling is invented. It is also refined (i.e. slightly reinvented) over time to better fit our discoveries. That, too, should be obvious to all but the most die-hard of determinists.
I know, this answer isn't very deep, but in my opinion the question isn't nearly as deep as it is being made out to be.
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Re:Logical positivism to the rescue... (Score:5, Funny)
Damn, I am too drunk to type. I have one eye closed as I type.... so you win
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Re:Logical positivism to the rescue... (Score:5, Informative)
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Re:Logical positivism to the rescue... (Score:5, Insightful)
Depends what you mean by "exists". For example, mathematical concepts are not observable (which is the condition for existence in an empirical framework), but physical systems can be observed which implement the concept. One can observe one apple or one galaxy, but one cannot observe the number one.
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Re:Logical positivism to the rescue... (Score:5, Insightful)
truth is discovered
truthiness is invented
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Re:Logical positivism to the rescue... (Score:5, Insightful)
Math is the symbolism used to describe the universe. Physical reality does not need symbols or tools or sentience to function, we however need math to describe the functions of the universe in precise detail. Math is a tool and so is an invented thing where the ideas have come from observing the world around us, just like a knife or velcro are tools that where invented based off of ideas gleaned from observations of the world around us.
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Re:Logical positivism to the rescue... (Score:5, Insightful)
No, the correct answer is "both."
No, I think the correct answer is, "What are you asking?"
The problem with questions like this is that it isn't clear what's in the mind of the person asking the question. What do you mean by "invented" and what do you mean by "discovered"? What difference do you see between the two?
For example, some people will think that "invented" means "made up". So in that person's mind, if math is "invented", then it's based only on human thought, and not on real principles of the universe itself. Of course, this line of thought makes me want to ask what it would mean to be a "real principle", and what is the "universe itself" when detached from human conception, but I'll leave that aside.
The problem I see immediately with this concept of "invented" is that real inventions don't exist independently of the universe. For example, was the wheel "invented", or did someone discover that rolling a circularly shaped object requires less energy than dragging an equally massive object? Was gunpowder "invented", or did someone discover than mixing certain chemicals together and setting fire to them caused an explosion? Was the telephone "invented", or did someone discover that you could convert sounds into electrical signals and back again by using magnets?
All inventions are a discovery of sorts, which makes this whole question a bit nonsensical.
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Re:Logical positivism to the rescue... (Score:4, Insightful)
No, I think the correct answer is, "What are you asking?"
.
.
For example, was the wheel "invented", or did someone discover that rolling a circularly shaped object requires less energy than dragging an equally massive object? Was gunpowder "invented", or did someone discover than mixing certain chemicals together and setting fire to them caused an explosion? Was the telephone "invented", or did someone discover that you could convert sounds into electrical signals and back again by using magnets?
Um, you have just given threee great examples supporting the original poster's answer of *both*...
In each case the basic scientific principle (mechanics, chemistry, elctricity & magnetism) was discovered (sometimes unwittingly) and then the knowledge of that discovery used to engineer an invention (wheel, gunpowder, telephone). The "discovery" was an observation of a natural phenomenom, etc, and the "invention" was creating something that otherwise did not exist in nature that took advantage of those phenomena. If you wanted to be pedantic you might argue the first "wheel" could have been discovered ("hey, look at how that round rock rolls!") but please don't try to claim that set of 18" forged alloy wheels with vulcanized radials was "discovered".
This is exactly the same argument the OP was making. Mathematics clearly involves the invention of a language to express discoveries (or assist in making those discoveries).
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Re:Logical positivism to the rescue... (Score:4, Interesting)
Semantics:
In old French, both were essentially synonymous.
inventer v.tr. To invent. (a) To find out, discover. [...]
Philosophy :
Under platonism, there's actually no distinction (see allegory of the cave).
By suggesting to let platonism die, the anonymous reader seems to want us answer "invent"...
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Score +1 (Score:5, Insightful)
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Re:Logical positivism to the rescue... (Score:5, Insightful)
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Re:Logical positivism to the rescue... (Score:5, Informative)
Check it out. cool stuff.
RS
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Re:Logical positivism to the rescue... (Score:5, Insightful)
Momentum scales linearly with both mass and velocity, fields fall off with inverse square relations, and so on. You cannot change the equations describing them away from these truths in any meaningful fashion without making the equations wrong - this is not human convention or definition, it is how the universe works.
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Re:Logical positivism to the rescue... (Score:5, Informative)
Your reasoning is subtly but fundamentally flawed. Yet as with all subtlties, pinpointing the exact nature of the flaw is difficult without having a back-and-forth conversation.
You are right on target with respect to Ohm's law and Hooke's law -- but quite off base with your general assertion. The deep laws of physics *are* eerily symmetric, independent of our need to describe them so.
For example, the inverse-square law of gravity or electromagnetism can be derived as a consequence of living in a 3-dimensional universe. (Integrate your favorite conserved quantity over concentric spherical surfaces and you get something that must "fan out" as 1/r^2). Nothing very suprising there. Nevertheless the deeper into exploration of physical laws you get, the more surprising interconnections pop up independent of our need to observe them.
Your assertion that "momentum" is simply a convenient and observed quantity is both false and misleading. "Momentum" is a fundamental quantity that relates directly and ... well, fundamentally to the nature of energy, space, time, et cetera. It is particularly noteworthy that the nature of space and momentum should relate to our perception of time -- a property/dimension/quality which is quite distinct from all others in its one-way observable nature. The laws of "physics" seem to be time-invariant, yet the laws of "thermodynamics" which are equally fundamental seem to recognize that time is somehow special.
Thus, it is misleading to imply that our physical laws are simple and elegant because we have simple and elegant requirements to describe the universe. An accurate description of the universe need not be simple -- and often it is not. For instance, I understand (although lack the mathematical sophistication to prove) that the electron spin g-factor has a theoretical value of exactly 2. Yet it is observed to be approximately 2.00232 and is one of the most precisely measured physical constants. So it is not always simple truth and beauty. Which makes it all the more surprising when the simplicity is there nevertheless.
And while it is true that the inverse-square law breaks down at relativistic energies, even that corrective factor of "gamma" remains mathematically simple, and in fact geometrically constructable via a pythagorean triangle analysis of a certain thought-experiment.
My point is that the easy examples are easily explained away by laymen, yet the surprisingly simple nature of the fundamental laws of the universe continue to pop up where you wouldn't expect. That is why expert scientists, true geniuses, of the sort that don't come along every day, routinely make comments about the "beauty" of physics. They have a deep understanding and "feeling" about the way the universe fits together that isn't captured by your example about momentum.
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Re:Logical positivism to the rescue... (Score:4, Informative)
What does this have to do with units?
Absolutely everything. Many fundamental equations of physics can be correctly arrived at simply by manipulating units. The dimensions of energy are kg*m^2*s^2. A combination of physical quantities which does not have precisely this dimension cannot possibly be a quantity of energy.
Dimensional analysis is an extremely powerful technique, and something which is learned in basic physics.
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Re:Logical positivism to the rescue... (Score:5, Informative)
Yes, it's also amazing that the equation isn't 2.14332544988e=2.14332544988mc^2.
Yes, sorry, I'm being a smart-ass and it's not polite. But c^2 is just a constant.
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Re:Logical positivism to the rescue... (Score:5, Insightful)
Other parts of math do resemble invention more than discovery. E.g., the definition of mole being the number of atoms of carbon 12 needed to make exactly 12g and the Coulomb, both of which are numbers that are arbitrarily assigned to fit in with the system of measurements that has been devised over the years. All of these constants could easily be multiplied by any non-integer value and the whole system would still work.
To answer the article's original question however, my answer would be: Who gives a toss? Math is useful. Whatever semantic definition we apply to the process by which we expand our mathematical capabilities has absolutely zero impact upon that expansion.
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Re:Logical positivism to the rescue... (Score:4, Interesting)
I deviated from the profession of mathematics long ago, but as far as I'm concerned, the question of invented/discovered was adequately retired by Kolmogorov-Chaitin complexity theory. For some reason, most mathematicians and most physicists seem determined to ignore this.
The formal system you begin with has an arbitrary beginning: the nature of the universal computer used to measure sequence length. In practice, the arbitrary starting point rarely makes a whiff of difference. The maximum disagreement on sequence length is bounded by the complexity of the program by which one machine is able to simulate the other. Since it is possible to construct universal computers of startlingly low complexity (you could easily write out the rules on the back of a business card with a blunt pencil), this bound tends to be minuscule for most universal machines we might choose to adopt for serious purposes.
I recall reading an article, by Putnam I think, where he talks about two different axiomatic formulations of the integers. Both formalisms agree on all the properties of the integers we regard as essential. However, in one system it is always true that if n < m then the set used to represent n id a subset of the set used to represent m. It the other axiomatic foundation, this is not true.
Some foundation points can introduce some strange discrepancies, but rarely anything we regard as material. This could probably be stated as an theorem in complexity theory. You'd have to put some elbow grease into the project to come up with a universal machine which can't compute pi using a "short" program where short is less than say Ackermann(4,4) and more likely, within a golf score of Ackermann(3,4).
Strange fact I didn't know:
http://en.wikipedia.org/wiki/Ackermann_function [wikipedia.org]
Perhaps this is why KC theory is so often ignored. People can't wrap their minds around A(4,4) as an example of an extremely small number. The problem is, the philosophical question of invented/discovered demands this cognitive shift. A(4,4) is *not* a large number on the *philosophical* landscape.
Chaitin's omega, however, is the total perspective vortex of theoretical mathematics.
There seems to be a small number of special constants, such as e and pi, that any universal computer anyone has ever found a use for can obtain from a short program. Within this nucleus, a nanoscopically small filagree in the multidimensional fractal of all possible mathematics, the balance shifts toward "discovered". The further one departs from this minute filigree of felicity and virtue, the more the scale tips toward "invented".
If that sounds like fluff, answer this: what is the shortest number one can copyright?
Due to subitization [wikipedia.org] it has never been possible to copyright the integers 1..4. The copyright on 5 probably expired 50,000 years ago. In modern society, there is evidence that 128-bit numbers remain fair game, though the difficulties of enforcing this are notorious. Clearly, five was discovered, the AACS constant was invented.
Not everyone agrees with Chaitin. This post makes a coherent statement of what he might be presuming:
http://coding.derkeiler.com/A [derkeiler.com]
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No, mc^2 is exact for an object at rest (Score:5, Informative)
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Re:No, mc^2 is exact for an object at rest (Score:5, Insightful)
This may seem like a nitpicking question, but it brings us to the point that I really want to make:
Mathematics is interesting because there are no ambiguities in a well described mathematical problem. There are many problems that have a finite set of solutions. However, every mathematical model we develop to describe our surroundings is only an approximation of our observations. With time, we can create more and more accurate models, but there will always be something about that model that is derived experimentally, and is therefor imperfect.
This does, in fact, tell us something about the underlying nature of the universe. Either it was created with some arbitrary parameters, or it exists in a way such that there is no way to perfectly describe it. Or maybe there are other possibilities I have not considered. What philosophical meaning you derive from all this is up to your own reasoning.
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Re:Logical positivism to the rescue... (Score:4, Funny)
"Hmm... I wonder if the larger box would still be better value for money if I were eating it in a spaceship with a velocity approaching c"
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Re:Logical positivism to the rescue... (Score:5, Insightful)
Suppose you had a definitive, 100% guaranteed answer to the "discovered vs invented" question. What would it allow you to do that you couldn't do before? What could you predict? What would you gain?
I tend to agree. I'm reminded of the Dutch computer scientist, Dijkstra, who said that ""The question of whether a computer can think is no more interesting than the question of whether a submarine can swim." Some questions are just meaningless.
I think the thing to learn here is that language isn't reality, it merely describes reality.
Oh, and the correct answer is "discovered"
No, I think the correct answer is "Why are you asking the question?" There might be a more interesting (and perhaps answerable) question that underlies it.
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Re:Logical positivism to the rescue... (Score:5, Funny)
Oh, and the correct answer is "discovered"
No, I think the correct answer is "Why are you asking the question?" There might be a more interesting (and perhaps answerable) question that underlies it.
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Re:Logical positivism to the rescue... (Score:4, Interesting)
Does mathematics which no one knows about exist?
Well it is obvious that on some level it should. It is likely that whatever new field of mathematics we invent, it will (eventually) be described using axiomatic set theory. But does the fact that we already have the language we need to describe a theorem mean that the theorem already exists? Does a sonnet exist before I write it? All the words I'm likely to use will be in some version of the Oxford English Dictionary. I can symbolically write down the abstract idea of every sonnet imaginable in only a few lines of mathematics. It would seem clear that mathematics, like poetry and prose, is invented then.
But then mathematics is different from prose, because mathematics can be used to make quantitative predictions about the world around us. It would seem that independent of human being nature itself 'knows' about mathematics. Before we invented calculus the acceleration on an electric charge due to a electromagnetic field could still be found using Maxwell's equations. The Falkland Islands were still there before the Spanish arrived right?
So now it would seem that at least some mathematics is discovered, at least as to how it relates to nature. Of course the mathematics we use to describe nature is just an approximation. Maybe nature doesn't know about math, maybe we just got luck.
Then there is another problem, whose to say that just because we think of prose as invented it really is. That might just be our sloppy use of language. I said earlier I could, at least in the abstract, write down every possible sonnet in the English language. That at least implies that those sonnets exist in some way before I write any of them, even if it is as an abstract sonnet.
Bottom line, it all comes down to what you think exists. If under your philosophy mathematical theorems can be shown to exist independent of if someone knows about them or not, then they are discovered. It is likely sonnet are discovered under that philosophy as well. If on the other hand theorems only exist after someone has conceived of them then they are invented. Now you have to be careful that at least some part of the Falklands weren't invented by the Spanish as well.
I'm going to have to go with discovered. To me Euler's equation is, was and ever shall be true and there isn't a darn thing anyone in this universe can do about it.
Of course the discussion doesn't really yield any useful results, so I would like to propose the Dirac interpretation for the uncovering of mathematical knowledge:
Shut up and calculate.
It comes with a corollary of my own devising:
No you cant patent it.
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Mathematics in the forms of human intuition (Score:5, Insightful)
I much prefer the Kantian approach, which, simplified, is that space and time are the forms of human intuition, and it is these forms of intuition that lead to us understanding things the way we do (spacially and temporally, whose relationships are mathematical). "Things in themselves" are unknowable, and can only be approached through some set of references, whether it be through the space and time we perceive, other possible ways time and space could work (non-Euclidian geometries?), or ways we can't even imagine. Unlike Plato's idea, which is that mathematics involves universal truths we discover, Kant's "Copernican turn" puts the subject as the one who projects mathematics onto everything it experiences. Arguably, this is the idea that has lead to the "modern era".
This makes mathematics the study of these forms of intuition, so unlike Plato's approach, we're not "discovering" universal ideas, but rather coming to understand the way we interpret the world (and by "we", I mean me, the beings who do science that makes sense to me, and probably most beings on earth whose methods of sensation resemble that of humans).
To answer the question of discovery or invention from this perspective, we can invent ways to do mathematics, but the relationships themselves are a discovery of the way we intuit anything we can sense.
I know this! (Score:5, Funny)
But did God invent or discover it? (Score:3, Interesting)
That would be a good question for Theists. The origin of the Universe poses few logical problems for a Theist (thousands of years ago thinkers realized the universe was a sub-reality like a story - or in modern computer terms, a virtual machine). But the origin of things like logic or justice are trickier. For instance, is everything God happens to do "good" because He is God and says so? That view is ca
Patently Obvious.... (Score:4, Insightful)
How is this a debate? It's both. (Score:4, Insightful)
What can be done with it is then discovered.
Re:How is this a debate? It's both. (Score:4, Insightful)
The concept was discovered, then we invented new methods of math based upon the discovery of Math
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Is Mathematics Discovered Or Invented? (Score:5, Interesting)
Neither. It is defined.
Re: (Score:3, Interesting)
Applied to math, you could say mathematics is a series of definitions we've created to describe an observed phenomenon or hypothesize the existence of an as yet unobserved phenomenon.
But what the hell do I know? I'm neither a philosopher or a mathematician.
Only the integers (Score:5, Interesting)
Integers were discovered. Beyond that, it's human invention.
I used to do work on mechanical theorem proving, and spent quite a bit of time using the Boyer-Moore theorem prover [utexas.edu]. When you try to mechanize the process, it's clearer what is discovered (and can be found by search algorithms) and what is made up. Boyer-Moore theory builds up mathematics from something close to the Peano axioms. [wikipedia.org] But it's a purely constructive system. There are no quantifiers, only recursive functions. It's possible to start with a minimal set of definitions and build up number theory and set theory. The system is initialized with a few definitions, and, one at a time, theorems are fed in. Each theorem, once proved, can be used in other theorems. After a few hundred theorems, most of number theory is defined.
But you never get real numbers that way. Integer, yes. Fractions, yes. Floating point numbers, representation limits and all, yes. But no reals. Reals require additional axioms.
Re:Only the integers (Score:4, Interesting)
Integers were discovered. Beyond that, it's human invention.
I can make a strong case that negative integers are invented. Because you can't have -3 apples. We invented the negative numbers to indicate a loss of positive numbers. We also invented fractions to stand in for ratios.
Sort of.
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Axioms vs. theorems (Score:5, Insightful)
The point is that the axioms don't exist until we create them. But once we create a set of axioms, then the results are an inevitable (if arduous) journey of discovery which might require clever inventions to reach the destination of mathematical knowledge.
both? (Score:4, Interesting)
Parallel (Score:5, Interesting)
Re:Parallel (Score:5, Funny)
- shazow
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All the same? (Score:4, Interesting)
Why so human-centric? (Score:3, Interesting)
So, regardless of the whole platonic debate, basic mathematics definitely exist independently of humans.
The super-imaginary number, j. (Score:3, Interesting)
You've just reinvented Projectively Extended Reals (Score:5, Interesting)
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What Erds and Feynman believed about this (Score:5, Informative)
Feynman also felt like coming up with a proof was more discovery than invention. He said that the proof felt like it was already there all along, raising the question of where "there" is.
Glib answer... (Score:4, Insightful)
To be more specific, Mathematical rules are discovered, Mathematical techniques are invented; "Mathematics" consists of both.
Just reading about this... (Score:5, Interesting)
Here's a thought problem for you.
You have the following in your hand:
A one-cent piece from 1978
A one-cent piece from 1986
A one-cent piece from 2004
I could have said you have 3 cents. But there is no such thing as 3 cents. 3 cents is an idea, an abstraction. It is not a concrete thing in the real world.
So, despite all that we appear to discover about the world through mathematics, we cannot really say that math is "out there" somewhere waiting for our discovery. Rather, mathematics is our projection onto the universe. It it because of the shortcomings of our abstractions and models that our science must be continuously revised.
For example, Newton did not discover anything about the universe. He made observations and rationalized (projected?) an abstract model which works very similarly to the observations. It's repeatable and consistent, so we call it a theory.
But then along comes Einstein. He makes some new observations, some new hypothesis, and voila, a new theory. Even if you argue that Einstein, or anyone else for that matter, has made such discoveries through mathematical observation, that doesn't discount the fact that the observation in that case is made upon the abstraction of the universe, not the universe itself.
In summary, mathematics is a simulation of the universe. It's an abstraction. One we humans invent. The fact that our model is observable, predictable, and so on in no way justifies the position that we are discovering some thing which pre-existed. Here's a final analogy - a computer model can be created to simulate the design of a car. We can study, observe, made predictions, corrections, and so on with the model. Yet, despite how relevant those observations, predictions, corrections, and so on are to the real car, they are still NOT the real car. The model is our interpretation, our abstraction of the car. We invent it. We make it. We project our ideas about the car into it. We do not "discover" it. The model does not exist without us.
Re:Lawyers are circling again I see (Score:4, Informative)
No. If it is invented it can be patented. If it is created it can be copyrighted. If it is discovered it can be neither patented nor copyrighted.
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Re: (Score:3, Informative)
Re:It's neither (Score:5, Insightful)
So you invent/assume your choice of axioms, and everything else follows from them and can be discovered at leisure.
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Re:Well it's obviously discovered (Score:4, Interesting)
I may be the nihilist, but you're the egotist - the one who believes that the order he sees in the universe is really there, not simply the result of his choice to define "order" in such a way that some parts of the universe seem to fit.
To suggest that we invent math is pompous at best.
To suggest that we discover it - that our brains, somehow, are able to tune in to an entire dimension of mystical mathematical truths - is arrogant.
And I have to ask you the question that completely dispels mathematical platonism - where do the wrong ideas come from? If they come from a special universe for wrong ideas, then discerning the difference is the same thing as inventing them. If they come from human imagination, if humans can invent wrong ideas, then surely they can invent right ideas too, and again, it's all invention.
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