The Question : is mathematics discovered or invented?

[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.