The Question : is mathematics discovered or invented?

[Q_Goest]

I may have misunderstood you and you may have already been saying the same thing I was with my chicken example. I was saying that simply because math could be formulated a different way doesn't make it an invention.

Yea, guess we both misunderstood. That post was replying to JoeDawg's post suggesting math is an invention. It is in fact, BECAUSE math may be realized in just about any way imaginable, yet the concepts can all be found to be "isomorphic" (if I may use one of your own favorite words) that mathematical concepts must have some basis which is independant of the language and therefore 'discovered'.

 

[Q_Goest]

I imagine, if we had an operational definition of the verbs 'to invent' and 'to discover', there wouldn't be any debate over the answer to the titular question.

If we allow everyone to color those words with their own personal biases, we have no hope of getting anywhere!

Rather than try to define the terms "discover" and "invented" I think a more useful thing to do would be to try and determine how many catagories the natural world might fit into. That's essentially what Penrose would like to do. He breaks the natural world up into the following three:

1. Physical world: Physical, 4 dimensional world. Includes mass, energy.
(Examples include protons, atoms, molecules, energy, cars, planets, people, galaxies, etc... )

2. Mental world: Not objectively measurable, so it doesn’t fit into the physical world.
(Examples include: the redness of an apple, the sweetness of sugar, love, hate, pain, etc...)

3. Platonic Mathematical world: Contains relationships that are 'perfect'.
(Examples include: mathematical operations (such as +, - and =), the mandelbrot set, ratios such as pi and e, etc...)

ok, the last one may or may not be an additional category, and I certainly haven't defined any of these very well, and I don't think my examples of the mathematical world are the best. <sigh>

But the question of "discover" or "invented" can be changed to one of catagorization. Are there more catagories than 3? Should there be less? Or is 3 and only 3 the perfect number?

(See also post #32 for further references to Penrose, Mazure and U of O)

 

[JoeDawg]

Okay then, if that's not math, what is it? What is math describing?

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The same thing english and russian are describing, things in the world. Some of those things appear to be objects, some are relationships between objects. Numbers and functions... Mathematics is simply more abstract and in some ways more precise, because we use it differently for different reasons.

 

[JoeDawg]

Hi JoeDawg,
What do you think of Penrose's suggestion:

... only a small part of the world of mathematics need have relevance to the workings of the physical world.

Only a small part of english need have relevance to the workings of the physical world.
We can talk of unicorns, gods, ETs, and Elves. We can talk about 'fiction' in english, we can even talk about 'nothingness', and some philosophers do, tediously. But these are merely imaginative recombinations of what is 'in the physical world'.

The point is, these are all different languages which describe the same thing, so I'm not sure we can say it is "invented" since others in distant reaches of the universe, could similarly come up with different mathematical languages which would describe the same mathematical concepts we have which, as Penrose notes, have no relevance to the workings of the physical world.

But that's equating the concept of something with the actual thing. A 'triangle' is no less an abstract object than a 'house' is. Both can exist in the real world, but never quite the way we can imagine them. All living creatures would likely need some sort of shelter, so across the universe, aliens probably have 'houses' or something like them.

Our imagination allows us to create many things not in the world. Why would mathematics be different from other languages. And just because 'the house" and "la maison" exist in the real world and describe similar things, doesn't mean that "El Chupacabra" and "the goat sucker" also do. I agree with what Penrose is saying about math, but I don't agree with his conclusion, which I feel is likely based on his love of math. Mathematicians are not the only ones to make this mistake, Philosophers have fallen into linguistic traps too, presuming that concepts in their language are universals in a separate reality. Its an old problem.

 

[Q_Goest]

Hi JoeDawg,
You make a very valid point. Thanks. Unfortunately, I think I've slightly misrepresented what Penrose is saying here. Or at least I've not been faithful to his argument.

Section 1.3 of his book is entiteld "Is Plato's mathematical world 'real'?" Penrose points out that math is robust such that others can verify mathematical truths. Mathematical truths can be agreed to (those parts external to nature that we talked about) regardless of cultural background. Note that this is not particularly true for language as you point out. We could describe various sciences, especially the soft sciences such as psychology or philosophy, in different ways just like blind men touching the proverbial elephant describing different parts. Or we can talk about Unicorns, ET's, elves, etc...

So when Penrose is saying, "... only a small part of the world of mathematics need have relevance to the workings of the physical world." he's not talking about something ficticious, he's talking about mathematical truths (he uses Fermat's Last Theorem as an example) that could be verified by alien life forms billions of light years away. Those same life forms could come up with the same fictious stories about unicorns or elves, but that isn't really the point Penrose is making. He's talking only about mathematical truths which have no relationship, no bearing whatsoever on the physical world.

 

[JoeDawg]

Hmmm. That does make the point clearer. Well, I'm still not sure I agree, but my level of math is rudimentary by comparison. I will say the theorem described strikes me more as a logic problem, or rather one describing the syntactical limits of the language of mathematics, ie given a certain starting point.

In an English sentence, we have an order of: subject(a), verb(b), object(c). Of course people fudge this all the time. By contrast though, mathematicians don't fudge, or consider it wrong if you fudge on agreed upon axioms or syntax.

If we and they were to be as strenuous with natural language however, an alien species would be just as limited as we are, in the ways they could say something... even if the language they spoke wasn't english, but rather a language similarly ordered. The rigid logical structure would rule. It would be inviolate.

Mathematics is much more precise and rigid, than other languages. Apply that to english and you would be accused of being an insane-grammar-nazi. But it empowers a level of logical thinking that natural languages simply can't touch.

In the end it may be my objection to Penrose is more about what he is indirectly implying, rather than what he is trying to say.

 

[CaptainQuasar]

The same thing english and russian are describing, things in the world. Some of those things appear to be objects, some are relationships between objects. Numbers and functions... Mathematics is simply more abstract and in some ways more precise, because we use it differently for different reasons.

Ah, but English and Русскии Язык may also describe things that are not in the world, as I said like Frodo Baggins or slapstick comedy. In those cases both the description and the thing described are invented, ньет? The question is whether mathematics is more like one of those things or if it's something more like gravity.

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[oldman]

But then, shouldn't you be willing to assert that every branch of science is also invented? Because a science never is what it describes...at least things like biology and chemistry and physics...The real question, I think, is whether it's invented in the same way that something like human culture is invented - “of whole cloth,” as it were

I've been going back over this thread, which is now getting a bit involved, and find it's clarified my thinking -- or at least modified my muddled mulling. Thanks for the help, folks.

In it's early stages people like Drachir (#2), Morodin (#4) and HallsofIvy (#5) claimed briefly that Mathematics is both discovered and invented. I now think there's some truth in this. Then Cap'n Q (#14) took the stance that Mathematics is entirely discovered, whereas my view (e.g. bolstered by Joedawg (#10) and Q_Goest's analysis (#32) had been that it's entirely invented. I now think that "entirely" is inappropriate; Cap'Q 's argument about geometrical ratios being discovered is persuasive.

Which brings me to this reply. It is certainly helpful to consider the nature of biology, chemistry and physics as well as mathematics, Cap'n Q. I would say that biology is pretty much entirely discovered, as are most aspects of chemistry. Physics seems to me substantially discovered -- invention (theory) and observation (discovery) have progressed hand in hand for a long while now. But now that its experimental frontiers have become inaccessibly extreme, physics includes large dollops of pure invention --- just look at string theory --- hard to deny that this is invention!

Cosmology, on the other hand, for a long time mostly invention, is finding it also has a robust supporting skeleton made of discoveries like the WMAP results. Just like old fishy Smith discovering the Coelocanth. And what about mathematics? Number theory and algebras seem to me proper examples of invented stuff. But there are aspects of geometry that seem to be discovered, like geometrical ratios (thanks, Cap'n), and statements like " the geometry of space sections is not Euclidean", lately found to be true (the images of distant galaxies are distorted by gravitational lensing). So I now accept that the early posts in this thread were correct -- mathematics is mostly invention, but also somehat discovered.

 

[CaptainQuasar]

And what about mathematics? Number theory and algebras seem to me proper examples of invented stuff. But there are aspects of geometry that seem to be discovered, like geometrical ratios (thanks, Cap'n), and statements like " the geometry of space sections is not Euclidean", lately found to be true (the images of distant galaxies are distorted by gravitational lensing). So I now accept that the early posts in this thread were correct -- mathematics is mostly invention, but also somehat discovered.

Ah! Now one other aspect of what I'm theorizing is that in the same way that set theory and graph theory are isomorphic, geometry and the other branches of mathematics are probably isomorphic to a deeper degree than humans can perceive. So I think that every postulate and deduction within number theory and algebras must also be expressible and provable in geometry as well!

(Though I don't think it's actually even geometry that is “real”, I think the real stuff is probably even more fundamental and something we might not even be able to recognize as akin to geometry.)

So even though number theory and algebra appear to humans to be expressing things that are too abstract to be directly connected to reality, they are in fact proving and specifying things that do identify congruences and structure in the real world.

But now that its experimental frontiers have become inaccessibly extreme, physics includes large dollops of pure invention --- just look at string theory --- hard to deny that this is invention!

I would agree with you that those parts of physics are invented since they must be trying to attribute untrue structures and properties to the real world. But the associated math isn't casting aspersions, it's simply determining self-consistent internal structure which, as I said, through isomorphism must be equivalent to something, somewhere in the real world. So I would say the math is actually more real than the physics!

Ha ha! Take that, physics!

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[oldman]

Ah! Now one other aspect of what I'm theorizing is that in the same way that set theory and graph theory are isomorphic, geometry and the other branches of mathematics are probably isomorphic to a deeper degree than humans can perceive. So I think that every postulate and deduction within number theory and algebras must also be expressible and provable in geometry as well!

You may be right here. Do you know "Geometrical methods of mathematical physics" by Bernard Schutz? He reinforces what you say. BUT -- has it struck you that, roughly speaking, geometry may well be real and to-be-discovered, while dialects like coordinate geometry, the ideas of manifolds etc. are invented to describe this reality, just as there seem to be inventions (languages, algebras) that describe this and other "realities".

Though I don't think it's actually even geometry that is “real”, I think the real stuff is probably even more fundamental and something we might not even be able to recognize as akin to geometry.

Maybe so, but I hope not. There are already enough mysteries to go around: QM for example.

So even though number theory and algebra appear to humans to be expressing things that are too abstract to be directly connected to reality, they are in fact proving and specifying things that do identify congruences and structure in the real world.

Yes, perhaps this is what I'm also suggesting.

I would agree with you that those parts of physics are invented since they must be trying to attribute untrue structures and properties to the real world. But the associated math isn't casting aspersions, it's simply determining self-consistent internal structure which, as I said, through isomorphism must be equivalent to something, somewhere in the real world. So I would say the math is actually more real than the physics!

In this case I agree strongly. There can be nothing wrong with the maths of string theory, but the physics looks very shaky, due to the lack of contact with prediction and verification. We are too given to building towering logical structures on foundations of sand, even when the mortar of logic which makes them cohere seems sound. For example the are a multitude of faiths that can't all be right. Yet they cohere, fiercely.

 

[CaptainQuasar]

You may be right here. Do you know "Geometrical methods of mathematical physics" by Bernard Schutz? He reinforces what you say.

I'm not familiar with it, I'll look it up.

BUT -- has it struck you that, roughly speaking, geometry may well be real and to-be-discovered, while dialects like coordinate geometry, the ideas of manifolds etc. are invented to describe this reality, just as there seem to be inventions (languages, algebras) that describe this and other "realities".

Yes, certainly. I pick geometry because its elements are less like pure symbols, as those of conventional algebra are, so it's more external. But they are still simply constructs to facilitate human thought, still intermediate to the real things they mimic. That's why I think there's probably something more fundamental that comprises the “real” stuff.

I think the concept of Platonic forms must have some truth or meaning to it, at least in the case of mathematics, because something like π is a commonality between many unconnected, disparate things. It seems like the “objects” of mathematics, like a circle or a vector field or a manifold, are really condensations of some diffuse generality, the way we sometimes speak of gravity as an “it” but other times speak in terms of “the law of gravity”. (And I would be saying that the diffuse generalities are what is more real whereas the Platonic condensations are an artifact of human understanding of it.)

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[CaptainQuasar]

I just had an interesting thought… it concerns the way I was saying that the circles which exist in reality as sections of spheres, as orbits, etc. aren't precisely geometric circles, aren't perfect circles, but instead have properties that converge upon those of a geometric circle? If superimposed upon one another the things in the real world which we would analyze with our mathematics would display a distribution of near-circular shapes around the locus, the center-line of a perfect circle.

That harkens to Plato's conception of the physical objects in a particular category being many shadows of a perfect Form that transcends them all. As though behind all of the chickens and mountains in the world there's a perfect über-chicken and a perfect über-mountain existent on a higher plane of reality.

Well, it just occurred to me that this relationship, of real thingie to über-thingie, is something of a parallel relationship to that of the modern QM “particle¹” that doesn't really have position or momentum and is rather a cloud of probability to the classical ballistic cueball-like particle that does have position and momentum.

I don't think this necessarily means anything profound. It's just an interesting connection I thought of.

¹ I put particle in quotes here because I think it was a poor choice for physics to not come up with a new term in the advent of QM. It seems to confuse lots of people particularly since the phrase particle/wave duality is still floating around.

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[oldman]

I just had an interesting thought…

I think this is the forum for such thoughts. I'll mull at yours.

Some time ago I had vaguely similar ideas, and posted https://www.physicsforums.com/showthread.php?t=124737" in the Quantum forum, which got exactly zero responses. Wrong forum, perhaps, or silly ideas. I think that folk who post in the Quantum forum are either baffled newbies to the subject, or busy and polite practitioners of QM (grad students?) too well-schooled in the subject to worry about its foundations.

In the meantime, remember that NOBODY, but nobody, yet truly understands QM.

 

[CaptainQuasar]

Another thing about QM is that in trying to firmly say something about its fundamental meaning it's very easy to make a statement that is easily disputed with evidence.

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[oldman]

... the things in the real world which we would analyze with our mathematics would display a distribution of near-circular shapes around the locus, the center-line of a perfect circle...this relationship... is something of a parallel relationship to that of the modern QM “particle¹” that doesn't really have position or momentum and is rather a cloud of probability to the classical ballistic cueball-like particle that does have position and momentum.


¹ I put particle in quotes here because I think it was a poor choice for physics to not come up with a new term in the advent of QM. It seems to confuse lots of people particularly since the phrase particle/wave duality is still floating around.

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You couldn't have touched on a better example of how a choice of an invented mathematical dialect (wave mechanics) has, in my opinion, resulted in endless confusion (particle/wave duality). in describing a quite simple phenomenon . See the https://www.physicsforums.com/showthread.php?t=124737" I referred to.

 

[Holocene]

Correct me if I'm wrong, but doesn't physics rely heavily on mathematics to accurately describe/analyze some very important principles?

If so, how can we claim to have any respectable handle on physics if our methods for analyzing it are merely "invented"?

 

[oldman]

Correct me if I'm wrong, but doesn't physics rely heavily on mathematics to accurately describe/analyze some very important principles?

If so, how can we claim to have any respectable handle on physics if our methods for analyzing it are merely "invented"?

You are quite correct about physics. It does rely on maths. Physics describes/analyses with language: sometimes it's an ordinary language like English. Often the description of a physics process/phenomenon also uses the language of mathematics.

Now all languages are invented things: think of the one you use and where it came from. Using maths makes the description quantitative and helps predict the future of the process. For instance one might want to explain or predict the colours of light emitted by an atom. To do this you use Quantum Mechanics, which may itself employ different mathematical dialects, like wave equations and wave functions, or operators, matrices and vector spaces. These dialects were also invented. Remember that a language is not the same as the things it describes. People tend to forget this.

 

[CaptainQuasar]

You couldn't have touched on a better example of how a choice of an invented mathematical dialect (wave mechanics) has, in my opinion, resulted in endless confusion (particle/wave duality). in describing a quite simple phenomenon . See the https://www.physicsforums.com/showthread.php?t=124737" I referred to.

Wave mechanics isn't a misnomer though, it really describes the mechanics of waves: ocean waves, sound waves, light waves, radio waves, the mechanical behavior of springs, ensemble phonon behavior in lattices, the slinky-like contractions and expansions in car spacing during traffic flow that civil engineers need to study for building highways and roads - anything that is cyclical or periodic.

The particle/wave duality was a genuine quandary in physics before the advent of QM. The terminological error was that after the advent of QM, once physicists knew that the itty-bitty things they were studying were neither ballistic particles nor waves in some medium, they should have come up with another term. But they didn't - they kept talking about particles, and nicknamed the Schrödinger equation the "wave function" because it's all curvy, even though trigonometric functions don't appear within it anywhere.

I think another thing that perpetuates the problem is, paradoxically enough, that the double slit experiment is so easy to perform. I remember doing it in public school when I was ten or eleven. And of course it's going to be performed at exactly the point you'd be studying the particle/wave duality, so of course that makes the subject stick more firmly in childrens' minds.

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[a2tha3]

Well in order to fully try to answer this question to the best of my ability, I think you should ask yourself and others whether anything is discovered or invented, and most likely you will get contradictory opinions. It's all a matter of perspective and in a religious perspective everything is invented (Just to be clear I'm agnostic) by 'God' and from a scientific perspective everything is discovered. Religion ultimately suggests that god is the source of everything, so that explains why people would pick invent, and science is all about discovery. The people who pick both (religious scientifics?) do so to attempt to extend their openess iin thier

 

[CaptainQuasar]

Well in order to fully try to answer this question to the best of my ability, I think you should ask yourself and others whether anything is discovered or invented, and most likely you will get contradictory opinions. It's all a matter of perspective and in a religious perspective everything is invented (Just to be clear I'm agnostic) by 'God' and from a scientific perspective everything is discovered. Religion ultimately suggests that god is the source of everything, so that explains why people would pick invent, and science is all about discovery. The people who pick both (religious scientifics?) do so to attempt to extend their openess iin thier

I'm an atheist myself but I will say that this is a completely false dichotomy. Religious and scientific are not opposites. Atheists who think that all of their thoughts derive from rationality are fooling themselves. I've met religious people who are far more rational than many other atheists I know.

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[a2tha3]

I'm an atheist myself but I will say that this is a completely false dichotomy. Religious and scientific are not opposites. Atheists who think that all of their thoughts derive from rationality are fooling themselves. I've met religious people who are far more rational than many other atheists I know.

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I got cut off in the middle of my post, and I am not even suggesting that religous and scientific are opposites, I'm only suggesting that religous people are going to lean towards invent (most if not all) and scientifics will probably lean towards discovery (most if not all)

Im not going to try to finish that post because I lost my strain of thought because of a brown out here, electricity can be unpredictable

I would also like to add I can be considered an atheist, and most of my thoughts come off of logic and rationality. Am I fooling myself by thinking that I think using logic and rationality? Are my cognitive skills somehow "magically" insufficient now?

 

[CaptainQuasar]

I got cut off in the middle of my post, and I am not even suggesting that religous and scientific are opposites, I'm only suggesting that religous people are going to lean towards invent (most if not all) and scientifics will probably lean towards discovery (most if not all)

I suppose if they thought the question was asking whether mathematics was invented by God, they might answer that way. Otherwise I don't see any reason why a religious person would take a particular side in this discussion - it seems to me as though you'd be making that suggestion based upon some sort of stereotype.

I would also like to add I can be considered an atheist, and most of my thoughts come off of logic and rationality. Am I fooling myself by thinking that I think using logic and rationality? Are my cognitive skills somehow "magically" insufficient now?

Did you miss the part where I said I'm an atheist? If you're saying that under your definition atheists can believe in magic, it doesn't do much for your claim on rationality of thought to be an atheist.

In my experience people who make a big deal of characterizing their own point of view as the logical and rational one, and someone else's point of view as illogical and irrational, rather than simply making points and arguments about particular topics, frequently aren't really so logical and rational upon close examination. Whether or not I categorize you in that group is, I hope, entirely dependent upon the degree of integrity you display in using those characterizations.

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[Pythagorean]

Correct me if I'm wrong, but doesn't physics rely heavily on mathematics to accurately describe/analyze some very important principles?

If so, how can we claim to have any respectable handle on physics if our methods for analyzing it are merely "invented"?

wait. Why wouldn't it be respectable just because it was 'invented'?

Toilet's and showers are invented. I think they're more respectable than some other options.

 

[oldman]

Wave mechanics isn't a misnomer though...
The particle/wave duality was a genuine quandary in physics before the advent of QM. ...
I think another thing that perpetuates the problem is, paradoxically enough, that the double slit experiment is so easy to perform. I remember doing it in public school when I was ten or eleven. And of course it's going to be performed at exactly the point you'd be studying the particle/wave duality, so of course that makes the subject stick more firmly in childrens' minds

Yes, I agree. People endow physical phenomena with the properties of the mathematical tools they use to describe them. In the case of very small-scale phenomena they forget that we are "just" trying to describe an unfamiliar milieu with mathematical dialects which were invented to describe macroscopic stuff, like ordinary waves. No wonder that there is confusion, some breakdown in congruence and alternative dialects, such as the Heisenberg formulation. QM is much less mysterious than it is sometimes made out to be. A very dangerous word to use is "is"; as in an electron "is" sometimes a wave, or it "is" a particle.

 

[oldman]

... Religion ultimately suggests that god is the source of everything, so that explains why people would pick invent, and science is all about discovery. The people who pick both (religious scientifics?) do so to attempt to extend their openess iin thier

You've got a point here that never crossed my mind, a2tha3. I've been amazed at the heat that the titular Question in this thread has raised, and the didactic fervour with which some folk defend the "discovered option". It may well be because the "invented" option carries with it religious overtones, or a legacy of such, even for both atheists and "religious scientifics". I'm just ignorant and uncaring, neither religious, atheistic nor agnostic. Not a "true scientist" either.

 

[Pythagorean]

I'm pretty sure I support the claim that mathematics is invented. The fact that their exists relationships between things like force and acceleration is the result of a discovery, but the language we've invented to express that relationship could have been designed a number of different ways.

 

[oldman]

I'm pretty sure I support the claim that mathematics is invented. The fact that their exists relationships between things like force and acceleration is the result of a discovery, but the language we've invented to express that relationship could have been designed a number of different ways.

Yes, I'm also pretty sure about this. Or rather so I thought , until Cap'n Q. raised doubts in my mind about geometry. When it comes to relationships that bear on shapes, like the ratio pi between a circle's circumference and diameter, I do get confused between 'invented' and 'discovered'.

But then I suppose one could draw any shape, perhaps a cartoon outline of a dog, and take the ratio of, say, the dog's perimeter to its nose-to-tail distance. One could then claim that this is an invented ratio, which is trivially invariant for all exactly similar shapes, and should be pitched into the 'invented' bin. All cartoons are invented!

Or is pi such a fundamental and universal ratio, integral to mathematics, that it must be regarded as some kind of eternal truth of the Platonic world that will always be there to be discovered, even long after the human race has committed some ultimate folly and perished?

And what about the Theorem you eponymously invented so long ago, Mr. Pythagorean? Or did you just discover it lying by the wayside?

 

[CaptainQuasar]

Or is pi such a fundamental and universal ratio, integral to mathematics, that it must be regarded as some kind of eternal truth of the Platonic world that will always be there to be discovered, even long after the human race has committed some ultimate folly and perished?

π shows up all over the place in trigonometry, which is of course fundamentally based upon the circle. And via trigonometry it is integral to wave mechanics. So I am inclined to think that it represents a deeper connection than just the circle itself.

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[Pythagorean]

Yes, I'm also pretty sure about this. Or rather so I thought , until Cap'n Q. raised doubts in my mind about geometry. When it comes to relationships that bear on shapes, like the ratio pi between a circle's circumference and diameter, I do get confused between 'invented' and 'discovered'.

But then I suppose one could draw any shape, perhaps a cartoon outline of a dog, and take the ratio of, say, the dog's perimeter to its nose-to-tail distance. One could then claim that this is an invented ratio, which is trivially invariant for all exactly similar shapes, and should be pitched into the 'invented' bin. All cartoons are invented!

Or is pi such a fundamental and universal ratio, integral to mathematics, that it must be regarded as some kind of eternal truth of the Platonic world that will always be there to be discovered, even long after the human race has committed some ultimate folly and perished?

And what about the Theorem you eponymously invented so long ago, Mr. Pythagorean? Or did you just discover it lying by the wayside?

well, pi is a number and not particularly mathematics. But as it were, pi is just a comparison (of a circle's radius to it's perimeter). As you said, you can compare any two things to discover their ratio, but you chose to use mathematics and numbers to express that ratio, a system invented (as I see it) by humans for reliability and accuracy.

Of course, as you see even with pi, it's accuracy is limited... nobody really knows that true value of pi, just a very good approximation of it. (Ah, approximation, another wonderful invention for when you don't need to be as accurate as math sometimes allows)

 

[oldman]

well, pi is a number and not particularly mathematics. But as it were, pi is just a comparison (of a circle's radius to it's perimeter). As you said, you can compare any two things to discover their ratio, but you chose to use mathematics and numbers to express that ratio, a system invented (as I see it) by humans for reliability and accuracy.

You have dispelled my last confusions, engendered by Cap'n Q. (I forgive him!), between discovered and invented. Thanks. I see more clearly now that 'circle' is an invented word that describes a particularly symmetric shape, approximated in the physical world for a variety of reasons, that can also be described with invented mathematical concepts like 'trignometric functions' or "intersections of a 'plane' with a 'sphere' ". And pi is an invented quantitative description of an attribute of this shape. All invented language, like the rest of mathematics, right through to Clifford algebras. Nothing discovered.

 

[CaptainQuasar]

well, pi is a number and not particularly mathematics. But as it were, pi is just a comparison (of a circle's radius to it's perimeter).

But circles and radii and perimeters… these aren't “real” things, to say that π is a comparison between things like comparing your true love to a summer's day. To be circular is a discovered common property of real things in the universe, as is to be wavelike - regularly periodic and cyclical so as to submit to wave mechanics analysis by engineers or physicists, or to be acidic, or to be oviparous.

For humans it's a more easily-grasped discovered property than the concept of a wave or an acid or oviparous reproduction but it's just as real. Perhaps there is some distant alien amoeboid race for whom oviparous reproduction and wave mechanics are learned in kindergarden and circles are the equivalent of quantum mechanics and rocket science. :shy: That's rather Gary Larson's Far Side http://img63.imageshack.us/img63/4605/farsideme0.jpg" …

As you said, you can compare any two things to discover their ratio, but you chose to use mathematics and numbers to express that ratio, a system invented (as I see it) by humans for reliability and accuracy.

Of course, as you see even with pi, it's accuracy is limited... nobody really knows that true value of pi, just a very good approximation of it. (Ah, approximation, another wonderful invention for when you don't need to be as accurate as math sometimes allows)

As I've agreed further up in this thread, mathematics as merely a set of descriptors is invented as is any descriptor and that aspect of any science. I think the question is, is the topic of mathematics more invented than are the topics of biology or chemistry or physics? Is circularity something that falls within the domain of one of those sciences, or does it fall within mathematics, or is it purely invented and not of the realm of reality?

We do know the true value of π, it's just that it's an irrational number and as such, rather than being expressed as a decimal or simple fraction must be expressed as something like

4 \sum_{n=0}^\infty \frac{(-1)^n}{2n + 1}

Unless, of course, you're saying that decimal numbers are true and mathematical limit expressions are not true.

I would say that mathematics enables very exact expression of uncertainty, rather than saying it allows precision or imprecision sometimes.

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[kmarinas86]

Mathematics was discovered when we learned how to give vocal callings to quantities. In order to do this, we had to identify objects by their qualia, and then give a sequence of words ascribed to the quantifying of these objects. Understanding of mathematics is derived from the tuning of our stimulus-response system. Mathematics was developed from the need to measure things.

 

[a2tha3]

Mathematics was discovered when we learned how to give vocal callings to quantities. In order to do this, we had to identify objects by their qualia, and then give a sequence of words ascribed to the quantifying of these objects. Understanding of mathematics is derived from the tuning of our stimulus-response system. Mathematics was developed from the need to measure things.

You can argue that it was invented by using this explanation... Replace "discovered" with "invented" and you have a comparable argument. I think that it is essentially impossible to determine how mathematics came to be, other than assuming that it was either discovered or invented. You can pick each one and come up with a pretty good argument and a pretty good counter-argument making it extremely difficult to come to a final conclusion.

If I had to pick one however, I would lean towards invention, because of primitive humans are more than likely capable of simple logic, thus were more than likely capable of doing math and inventing math.

 

[oldman]

Mathematics was discovered when we learned how to give vocal callings to quantities. In order to do this, we had to identify objects by their qualia, and then give a sequence of words ascribed to the quantifying of these objects. Understanding of mathematics is derived from the tuning of our stimulus-response system. Mathematics was developed from the need to measure things.

You say that mathematics was discovered, and then go on to explain carefully how it was invented. Did you notice the title of this thread, kmarinas86?

 

[CaptainQuasar]

I think that it is essentially impossible to determine how mathematics came to be, other than assuming that it was either discovered or invented. You can pick each one and come up with a pretty good argument and a pretty good counter-argument making it extremely difficult to come to a final conclusion.

I think a pretty good approach is exactly the one you took with kmarinas86 there: to examine the definition of “discovered” and “invented” and see whether via a given proposed definition and set of arguments everything in science and scholarship turns out to either be completely discovered or completely invented.

In my opinion arguments like that - that everything is discovered and not even admitting that language and description are human-authored devices, or that everything is invented and acting as if there isn't the slightest external influence involved at some point, must be dealing with a fairly mundane and tautological definition of the terms involved. I guess that's the degree I'm willing to concede to the “both discovered and invented!” crowd.

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[Pythagorean]

But circles and radii and perimeters… these aren't “real” things, to say that π is a comparison between things like comparing your true love to a summer's day. To be circular is a discovered common property of real things in the universe, as is to be wavelike - regularly periodic and cyclical so as to submit to wave mechanics analysis by engineers or physicists, or to be acidic, or to be oviparous.

For humans it's a more easily-grasped discovered property than the concept of a wave or an acid or oviparous reproduction but it's just as real. Perhaps there is some distant alien amoeboid race for whom oviparous reproduction and wave mechanics are learned in kindergarden and circles are the equivalent of quantum mechanics and rocket science. :shy: That's rather Gary Larson's Far Side http://img63.imageshack.us/img63/4605/farsideme0.jpg" …

As I've agreed further up in this thread, mathematics as merely a set of descriptors is invented as is any descriptor and that aspect of any science. I think the question is, is the topic of mathematics more invented than are the topics of biology or chemistry or physics? Is circularity something that falls within the domain of one of those sciences, or does it fall within mathematics, or is it purely invented and not of the realm of reality?

We do know the true value of π, it's just that it's an irrational number and as such, rather than being expressed as a decimal or simple fraction must be expressed as something like

4 \sum_{n=0}^\infty \frac{(-1)^n}{2n + 1}

Unless, of course, you're saying that decimal numbers are true and mathematical limit expressions are not true.

I would say that mathematics enables very exact expression of uncertainty, rather than saying it allows precision or imprecision sometimes.

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Circles:

I think circles are very real, personally. I also believe they are a human invention, and that pi is a sort of result of the analysis of this invention.

Expression of Pi:

Even the sum form of pi "approaches and asymptote" in human understanding as it goes to infinite. After a lot of physics classes, I can sort of feel out 100/.001 as infinite. But I have no real feeling for infinite itself or 1/0.

 

[fuzzyfelt]

I would say that mathematics enables very exact expression of uncertainty, rather than saying it allows precision or imprecision sometimes.

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Nice thought. Nice rectangle, too.

 

[CaptainQuasar]

Circles:

I think circles are very real, personally. I also believe they are a human invention, and that pi is a sort of result of the analysis of this invention.

So if there were no people, nothing in the universe would be circular? That ought to be the consequence if circles are merely a human invention.

Expression of Pi:

Even the sum form of pi "approaches and asymptote" in human understanding as it goes to infinite. After a lot of physics classes, I can sort of feel out 100/.001 as infinite. But I have no real feeling for infinite itself or 1/0.

I think you misunderstand a few of these concepts here. A limit is not the same thing as an asymptote. There's no change, it doesn't approach anything, it's a static value. The value of the above expression is π, it's not an approximation, you could replace π in any expression with it and manipulate everything per mathematical rules and get exact answers, not approximate ones. (But again, exact answers that are expressions containing limits, which are just as true as numbers.)

Infinity in mathematics is not a number, it's not even a specifically define object, it's a general concept (in conventional mathematics). 100/.001 is not infinite, nor is any other number. 1/0 is undefined, not an infinite value.

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[Q_Goest]

To Penrose's point:
1. Library of fiction created by authors on Earth.
2. Library of mathematics created by authors on Earth.

The first of these is highly unlikely to be found anywhere except on Earth.
The second might be found in every advanced civilization throughout the universe, albeit, written in a slightly different mathematical language.

If I wanted to find a book on Fermat's Last Theorem in the Andromeda Galaxy on a planet where snortblots have three toes and thus have a base 6 numbering system, I should be able to find it.

Conclusion: Mathematics is discovered.

 

[JoeDawg]

Conclusion: Mathematics is discovered.

That conclusion is based on a false analogy. One could easily find a book on linguistics in the library of any species... with language. Even if the language was suitably different. Math isn't just about measurements, but also relationships between measurements.

 

[Q_Goest]

Why would linguistics necessarily be the same for different developed languages? The German language for example, often puts the verb at the end of the sentence. Languages don't have the same structure. We may find they all have some kind of structure, but because they have different structures to perform the same task, we must then conclude those structures are invented. - Hope I haven't misinterpreted what you mean.

Math isn't just about measurements, but also relationships between measurements.

That's actually the point Penrose is making. Math isn't just about measurements and relationships in the physical world. He states that math has only some limited application to the physical world, and therefore, since the rest of the math (such as Fermat's last theorem) is real and could be found anywhere in the universe (ie: by any alien species) then it's discovered.

Note, Penrose actually uses the concept of "mathematical world" as opposed to "discovered/invented" but the meaning is the same.

 

[oldman]

To Penrose's point:
1. Library of fiction created by authors on Earth.
2. Library of mathematics created by authors on Earth.

The first of these is highly unlikely to be found anywhere except on Earth.
The second might be found in every advanced civilization throughout the universe, albeit, written in a slightly different mathematical language.

If I wanted to find a book on Fermat's Last Theorem in the Andromeda Galaxy on a planet where snortblots have three toes and thus have a base 6 numbering system, I should be able to find it.

Conclusion: Mathematics is discovered.

In this post it seems to me that you are indulging in wishful thinking. If the Platonic world existed, and if mathematics were part of it, you might indeed be able to find a book on Fermat's last theorem in the Andromeda Galaxy -- if an advanced book-writing civilization existed there. But you don't in fact know that any of these things are true. You are simply assuming your conclusion. And much else.

 

[JoeDawg]

Why would linguistics necessarily be the same for different developed languages?

Seems to be the case.

The German language for example, often puts the verb at the end of the sentence.

A better example would have been something like Chinese or ancient Egyptian or American Sign Language. If you were going to emphasize differences in language. German... compared to English certainly has differences, but the overall structure is quite similar. Both have verbs for instance.

Does it really matter if one says 2 x 3 = 6, rather than 3 x 2 = 6?
Is one saying something quantitative different if binary was used? How about Hex?

Not really. Of course if you don't know binary...it wouldn't make a lick of sense.

Note, Penrose actually uses the concept of "mathematical world" as opposed to "discovered/invented" but the meaning is the same.

'Imaginary world' works for me.

 

[oldman]

So if there were no people, nothing in the universe would be circular? That ought to be the consequence if circles are merely a human invention.

I don't quite see why Pythagorean said "I think circles are very real, personally" while believing that they are invented. Maybe one jumped out and bit him on the leg while he, personally, was walking in the woods? I prefer to think this was just a slip of his pen.

But your reply is nearly as obscure. If circles were invented, and the people who had invented them vanished (so that there were no people, as you propose), then of course there would be no circles. But the old universe would roll along, as it were, replete with all the things that had shapes the vanished people had described as circular. You fail to distinguish between objects that have (approximately) circular shapes and the idealised human concept of a circle -- just a word to describe with.

By the way, I've tracked down the villain who started all this argument by inventing NUMBERS. He was a Greek actor, name of Palamedes, who "claims to have invented number" and "counted the ships and everything else" that sailed to Troy. This "implies that nothing had been counted before and that (King) Agamemnon, apparently, did not know how many feet he had". I have this on the best philosophical authority, as related by Plato in his dialogue "Education of the Philosopher". I now suspect that even modern phiosophers do need educating!

 

[Pythagorean]

So if there were no people, nothing in the universe would be circular? That ought to be the consequence if circles are merely a human invention.



I think you misunderstand a few of these concepts here. A limit is not the same thing as an asymptote. There's no change, it doesn't approach anything, it's a static value. The value of the above expression is π, it's not an approximation, you could replace π in any expression with it and manipulate everything per mathematical rules and get exact answers, not approximate ones. (But again, exact answers that are expressions containing limits, which are just as true as numbers.)

Infinity in mathematics is not a number, it's not even a specifically define object, it's a general concept (in conventional mathematics). 100/.001 is not infinite, nor is any other number. 1/0 is undefined, not an infinite value.

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Circles:

I disagree that circles being a human invention ---> circular things only exist because of humans.

Circular things are NOT circles. We can only describe "circular things" because we invented a scale (the circle) to compare it to.

Just like an inch. Yes, I do really have a four inch thumb. I discovered that my thumb has four inches, but the inch I'm comparing it to I made up.

The Math


You misunderstood me. I probably shouldn't have used that metaphor in this discussion. I was trying to be punny, sorry. I was comparing the incomplete decimal set of the decimal form to the infinite sum in the sum form, not mathematically, but in terms of human understanding. When I said "approaches an asymptote" if you'll reread it, I was referring to my understanding not quite grasping the concept of infinite. I have a general idea of it.

I know 100/.01 isn't really infinite, but that's the only practical time the concept of infinite enters into physics discussion (it's been replaced with "blowing up" by some professors) and not specifically that quantity, of course... I'm merely illustrating how infinite applies to our every day experiences (which it doesn't without some approximation as I have done... because... perhaps infinite is a human construct?)

 

[CaptainQuasar]

I think Q_Goest's point is excellent: the reason that understanding of mathematics would be at least partially the same between humans and a distant alien civilization is because both civilizations will have discovered and been studying the same things. And by the way, the basic things that all human languages have in common are at their fundament mathematical: http://mathworld.wolfram.com/Grammar.html" , for example, is a genius linguist (and a genius in many other rights) who is skilled at the mathematical expression and analysis of language.

In this post it seems to me that you are indulging in wishful thinking. If the Platonic world existed, and if mathematics were part of it, you might indeed be able to find a book on Fermat's last theorem in the Andromeda Galaxy -- if an advanced book-writing civilization existed there. But you don't in fact know that any of these things are true. You are simply assuming your conclusion. And much else.

No more so than you guys are assuming your own conclusions. Were a distance alien civilization to exist, would they be aware of gravity? And if they were, would they be aware of π? And periodic functions and wave mechanics?

As I said, these things might be more peripheral to their mathematics-equivalent compared to ours. But they would not arrive at anything contradictory to human mathematics, the same way they wouldn't decide that gravity is a force that repels mass away from other mass. They wouldn't decide that the ratio of a circle to its diameter is exactly 4.07778 - except insofar as oldman points out that they might have a different basis for their geometry, in which case a Lorentz-transformation-like metric adjustment would demonstrate the same value of π as us - whether they're deriving that value from circles or from the oscillation of vibrating particles.

JoeDawg uses the even simpler example of “Does it really matter if one says 2 x 3 = 6, rather than 3 x 2 = 6?” Aliens are not going to conclude that 2 x 3 = 7 or 3 x 2 = 7, nor that commutivity is invalid in real number multiplication. Just the same way that you haven't had those sorts of differing conclusions amongst human cultures who progressed in mathematics independently.

But your reply is nearly as obscure. If circles were invented, and the people who had invented them vanished (so that there were no people, as you propose), then of course there would be no circles. But the old universe would roll along, as it were, replete with all the things that had shapes the vanished people had described as circular. You fail to distinguish between objects that have (approximately) circular shapes and the idealised human concept of a circle -- just a word to describe with.

No, I definitely have made the distinction in the course of this thread. Remember how I said that the circular things in the universe, taken in total, will like a mathematical limit approach the form of a circle / arrangement equidistant from a single point?

Were humans to vanish, the congruences and behavior in the real world described by our mathematics would continue and all of the properties we attribute to physical objects based on our mathematical analysis of them would continue to hold. Circular objects would still be more likely to roll downhill than square ones, for example. Frequency and wavelength in a wave or other regularly oscillating phenomenon would still be related by v=λf.

Circles:

I disagree that circles being a human invention ---> circular things only exist because of humans.

Circular things are NOT circles. We can only describe "circular things" because we invented a scale (the circle) to compare it to.

Just like an inch. Yes, I do really have a four inch thumb. I discovered that my thumb has four inches, but the inch I'm comparing it to I made up.

For your sake, I hope your thumb is something that you discovered and really exist rather than something you've invented out of thin air.

Thumbs are not thumbs. We can only describe thumbs because we invented a perfect thumb to compare them to.

I know 100/.01 isn't really infinite, but that's the only practical time the concept of infinite enters into physics discussion (it's been replaced with "blowing up" by some professors) and not specifically that quantity, of course... I'm merely illustrating how infinite applies to our every day experiences (which it doesn't without some approximation as I have done... because... perhaps infinite is a human construct?)

Let me know when you hit http://en.wikipedia.org/wiki/Taylor_series#List_of_Taylor_series_of_some_common_functions". Almost any quantity you use in any of your calculations is equivalent to an infinite geometric series. You're working down at an end of mathematics where things have been neatly nipped and tucked to iron out pesky and confusing infinities, but they're all around you. Like The Matrix, you just don't see them. But they're just as true and legit as the neat, packaged integers and real numbers that are easier to use.

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[Q_Goest]

If the Platonic world existed, and if mathematics were part of it, you might indeed be able to find a book on Fermat's last theorem in the Andromeda Galaxy -- if an advanced book-writing civilization existed there.

This is a nit pick. I'm not arguing that there will be a world with intelligent life outside our solar system. I'm not arguing that particular intelligent life form will necessarily find Fermat's last theorem. I'm saying it should be intuitively obvious that given the proper circumstances, Fermat's last theorem can be derived by any intellegence, independant of whereabouts in the universe, just as 1+1=2 and the ratio of diameter to circumference equals pi. These are truths which can be found which are independant of numbering system used, independant of culture, independent of location in the universe, etc...

The above is untrue of linguistics - which is invented. Similarly, the sciences in the format they are in on Earth can be rewritten in different ways.

Does it really matter if one says 2 x 3 = 6, rather than 3 x 2 = 6?
Is one saying something quantitative different if binary was used? How about Hex?

Not really. Of course if you don't know binary...it wouldn't make a lick of sense.

What you're stating is self evident. These are mathematical laws (commutative law) which hold true and can be found to be true regardless of where in the universe you are born. All intellegent life that discovers laws of mathematics should, in principal, be able to verify that the commutative property is true.

The point is that truths such as the verb has to go at the end of the sentance are not like mathematical laws (ex: commutative law) at all. They only hold true for that particular language.

Perhaps you are arguing that verbs and nouns can equally be said to be discovered. If so, I like that view. It may be a valid argument which needs to be explored. All the more reason why the original question regarding "discovered or invented" is erroneous (as I've tried to argue previously) and should be changed to a catagorical question. If the OP is changed to a catagorical question instead, I think the verb/noun issue will fall out into mental and physical worlds and Penrose's view will hold.

 

[Moridin]

The platonic circle is an idealized conception. The universe has no problem with wonky circles.

 

[Pythagorean]

For your sake, I hope your thumb is something that you discovered and really exist rather than something you've invented out of thin air.

Thumbs are not thumbs. We can only describe thumbs because we invented a perfect thumb to compare them to.

I think we agree here? Unless you're being sarcastic over the internet. Which is fail.

Let me know when you hit http://en.wikipedia.org/wiki/Taylor_series#List_of_Taylor_series_of_some_common_functions". Almost any quantity you use in any of your calculations is equivalent to an infinite geometric series. You're working down at an end of mathematics where things have been neatly nipped and tucked to iron out pesky and confusing infinities, but they're all around you. Like The Matrix, you just don't see them. But they're just as true and legit as the neat, packaged integers and real numbers that are easier to use.

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Perhaps you're still misunderstanding me. I don't think my understanding of inifnite is poor as a person, I think you me, and everyone have a poor understanding of infinite, and to some extent it's up to our imagination to understand it conceptually. I mean this in comparison to things we can tangibly relate too.

I'm in my last year of physics classes, but I'm thinking of going back and double-majoring in math too. I have definitely seen the Taylor series... of course... truncated... so that it's exactly back to what I was saying before (because you'd never solve a problem if you didn't truncate, obviously).

Once we truncate, we're back to the same discussion of approximation. If you don't truncate, and you actually want to know real values for a real system, then you're going to be calculating for a long time (infinite time, I'd assume).

Of course, there are special cases, like geometric series where we know that infinite sum results in a finite number, but it's not like we actually go out to infinite with the index; we derive a shortcut formula. My point is that we never actually experience infinite.

 

[JoeDawg]

The point is that truths such as the verb has to go at the end of the sentance are not like mathematical laws (ex: commutative law) at all. They only hold true for that particular language.

A verb has a specific function. Where you put it depends on the logic of your particular system. But its function is constant. Given the same experience with the world... the logic of the position becomes obvious. The fact is, natural languages aren't as rigid as mathematics. But that doesn't make math any less created by humans.

Observations of reality are what is real. We then create models... words...phrases... equations... to describe that reality. If we describe it well... then we can use that description to predict what will happen in reality. Math is artificial in every sense, its useful because its a rigid system but there are plenty of equations that don't describe reality, and those exist because we have creative minds, not because they exist separate from us.