The Question : is mathematics discovered or invented?


by oldman
Tags: discovered, invented, mathematics
oldman
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#19
Feb15-08, 04:46 AM
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Quote Quote by CaptainQuasar View Post
However anthropocentric mathematics is there's something “real” that is isomorphic to it and embedded in the universe.
Yes, there must be; I agree strongly. Otherwise we couldn't use mathematics as a tool with which to construct physics or build aircraft. It's not like ordinary malleable languages, French or English, say, that change substantially as time passes.

I suspect that both the Physical world and the language of mathematics share the same quality in that they are both rigidly logical. Neither are in the least magical or supernatural. Of course there are dialects of mathematics that fail as tools. Perhaps Quaternions? Or the mathematical development of string theory? So mathematics could be a more general structure than reality. Just speculating!.
oldman
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#20
Feb15-08, 04:54 AM
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Quote Quote by JoeDawg View Post
A number system that doesn't describe reality is discarded. Its wrong to say 1+1=3, because that doesn't describe reality.
Or it is not a "number sytem" at all because its elements do not qualify as the elements of a group, in the mathematical sense? I'm not sure about this reason, though.
HallsofIvy
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#21
Feb15-08, 09:04 AM
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Quote Quote by oldman View Post
Yes, there must be; I agree strongly. Otherwise we couldn't use mathematics as a tool with which to construct physics or build aircraft. It's not like ordinary malleable languages, French or English, say, that change substantially as time passes.

I suspect that both the Physical world and the language of mathematics share the same quality in that they are both rigidly logical. Neither are in the least magical or supernatural. Of course there are dialects of mathematics that fail as tools. Perhaps Quaternions? Or the mathematical development of string theory? So mathematics could be a more general structure than reality. Just speculating!.
I disagree that there must be some philosophical (mystical?) connection between mathematics and the "physical world". Since I just posted on another thread about that, I'll just give the link:
http://www.physicsforums.com/showthread.php?t=215462
Lord Ping
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#22
Feb15-08, 02:24 PM
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I'm just going to float this. Sorry if it seems irrelevant.

There is a type of cicada that comes out of the Earth in swarms every 17 years. The explanation for this is that, over evolutionary time, it's been battling with a predator that also emerged periodically. The predator couldn't get its periodicity exactly the same, but when the two periods hit a common multiple the predator would decimate the cicadas. So they evolved a cycle lasting a prime number of years, to minimise the chance of the predator emerging at the same time.

Great explanation isn't it? But it rests on the cycle having a particular real attribute: the attribute of being a prime number of years long.

If we weren't here, it would still have that attribute. So primeness is a real attribute out there independently of us. So we discover prime numbers, we don't invent them. Discuss...
CaptainQuasar
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#23
Feb15-08, 05:24 PM
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Quote Quote by HallsofIvy View Post
I disagree that there must be some philosophical (mystical?) connection between mathematics and the "physical world".
Saying that there is an isomorphism between something in the real world and human mathematics, which is the statement that oldman was responding to, is not proposing something mystical or philosophical - it's proposing a mathematical connection.

From the other thread you linked to:

Quote Quote by HallsofIvy View Post
Since real problems always involve measurement, which is approximate, we cannot expect them to be exactly true.
No, they are true in that they are like mathematical limits for the equivalent phenomena in the real world. Like I said, the ratio between the diameter and circumference of circle-like things converges on something like π.
oldman
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#24
Feb15-08, 11:03 PM
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Quote Quote by HallsofIvy View Post
I disagree that there must be some philosophical (mystical?) connection between mathematics and the "physical world".
I strongly concur.

The nature of the link between the language of mathematics and the the physical world -- the reason why mathematics is so effective in describing this world -- while not well understood, will, I think, turn out to be not in the least philosophical or mystical.

It's a practical matter that needs clarification, which I hope the thread you kindly linked to will provide. I reckon the reason could be rooted in the absolutely non-mystical and totally logical character shared by mathematics and the structure of the physical world it is used to describe. Both seem uniquely free of the plague of nonsense which so infests most human discourse.
oldman
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#25
Feb15-08, 11:24 PM
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Quote Quote by Lord Ping View Post
.... So primeness is a real attribute out there independently of us. So we discover prime numbers, we don't invent them. Discuss...
Or do we invent numbers, and then invent a set of things thing called a group that numbers are members of, and then invent a category of "prime" into which some numbers fall. A subgroup? Perhaps this is the way mathematics evolves into a network of logical inventions. But I'm no mathematician, and I don't know much about group theory, so I'm probably lost here!
LightbulbSun
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#26
Feb16-08, 12:37 AM
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Quote Quote by oldman View Post
Thanks for the reply, Drachir. Yes, but was mathematics "previously unknown" and "previously unseen", and therefore previously existing, waiting to be discovered. Or was it invented? I don't quite see how you drew your conclusion.

It may be helpful to start at the beginning, which for mathematics was simple arithmetic,
perhaps that of ancient folk like the Sumerians. Algebra came later from the same part of the world.

I, for one, tend to be smitten with too much awe by modern mathematical architecture, embellished with differentiable manifolds and fiber bundles, to think coherently about its structure and the axioms in its foundations. A question I find simpler but relevant is: Was arithmetic invented or discovered? Were the counting numbers discovered or invented? Or were these abstractions dormant entities in an abstract world waiting to be discovered once we had evolved from worms and worse?
Math is a mental construct. You don't walk out in the woods and find a number 4 roaming about. A number is a mental construct which quantifies our surroundings.
oldman
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#27
Feb16-08, 01:00 AM
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Quote Quote by LightbulbSun View Post
Math is a mental construct. You don't walk out in the woods and find a number 4 roaming about. A number is a mental construct which quantifies our surroundings.
Yes, I've not yet found the woods to be fraught with roaming numbers. Well said. And you invent metal constructs rather than discover them.
CaptainQuasar
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#28
Feb16-08, 01:08 AM
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Quote Quote by oldman View Post
Both seem uniquely free of the plague of nonsense which so infests most human discourse.
Gumple frubble dinken bop, arf garf woolay?

Quote Quote by LightbulbSun View Post
Math is a mental construct. You don't walk out in the woods and find a number 4 roaming about. A number is a mental construct which quantifies our surroundings.
I agree with what you're saying about numbers, but take the geometric ratio “quadruple”. You would find geometric proportions in reality at least as converging limits as I explained above. That's why while I think numbers are symbolism or a helper construct employed by humans, geometric proportions or something like them are “real”.
oldman
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#29
Feb16-08, 01:35 AM
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Quote Quote by CaptainQuasar View Post
........geometric proportions or something like them are “real”.
I think you're probably right. Are shapes like circles "real" or just patterns we pattern-recognising animals pick on and label? I can't decide.
CaptainQuasar
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#30
Feb16-08, 01:47 AM
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I don't think “pattern” and “real” are incompatible. Light is a cyclical oscillation of electrical and magnetic fields - does that make it “just a pattern” and not real?

Have you ever fiddled around with an implementation of “Conway's Game of Life”? It does a good job of demonstrating how patterns can have stability and properties of their own. Stephen Wolfram (of Wolfram Research which makes the tool Mathmatica and supports the Mathworld web site) certainly thinks they do, he thought he was going to be able to rewrite all of science in every field with cellular automata in his arrogantly-named A New Kind Of Science.
oldman
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#31
Feb16-08, 06:13 AM
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Quote Quote by CaptainQuasar View Post
(1) Light is a cyclical oscillation of electrical and magnetic fields - does that make it “just a pattern” and not real? ....

(2) ....Have you ever fiddled around with an implementation of Conway's Game of Life[/URL]”?
(1) Herein lies a can of worms. The concept of a "field" (invented by Faraday?) is quintessentially abstract, and has proved endlessly useful in physics, right up to the idea of the inflaton field conjured out of thin air by cosmologists who promote the inflationary scenario. I must confess that I've used the concept myself, without much thought. But I'm pretty sure fields are not part of the "real" world -- just an very useful description of one of its aspects. So I'd say that patterns of fields are like mathematics -- invented rather than discovered.

(2) No, I haven't -- but thanks for the URL's. I've been aware of the structures created by this method, which I've classified (rather arbitrarily) as being the result of the clever "tricks" (“organizational attractors” to you!) devised by Conway for the lattice of cells he invented and that are played with. I would descibe this Game of Life as fooling around with structures that evolve --- but others in this forum would consider my use of the word "evolve" cavalier. And I've not read Wolram's book; only heard of it.
Q_Goest
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#32
Feb16-08, 07:02 AM
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Hi oldman,
So do you really live something like 100 km NW of Durban? Must be interesting. I’ve had two cousins live in South Africa but never had a chance to visit myself.

You have a nice writing style too. I enjoy your symbolism.

I’d agree Penrose would opt for, or perhaps more appropriately would be adamant about, “discovered”. One can’t deny he’s one of the most brilliant mathematicians in the world so rather than throw my idiotic 2 cents in, I’ll look to see what Penrose has to say. Funny also that Mazur, although writing a paper that tries to portray the two sides without too much bias, also seems to be a Platonist. Or at least refuses to accept the anti-Platonist view.

The problem with the question however, is that it’s just too short. And the paper by Mazur, although spirited, doesn’t seem to really explain very well what is meant by “discover” and “invent”. Instead, his paper seems to assume you already know what the argument is all about. So I apologize for the length of this post, but I think we have to understand what is meant. For that I’ll digress momentarily and come around to try and explain my understanding of Penrose’s view, because I think it’s Penrose that really fleshes some of this out nicely.

Here, I’ll treat the word “physical” to mean that which can be objectively measured and found to exist in 3 dimensions and that of time. In this sense, something which is physical is a subset of the natural world since there are other phenomena which exist that can’t be considered physical. <gasp! more in a moment..> So I’ll consider the word “natural” to mean everything which exists that is both objectively observed and subjectively observed.

- For the natural world, discovered means that which existed at all times.
- Invented means that which came into existence only because of happenstance.

This is a slightly different definition of the terms than might be used elsewhere so I’ll try and explain what is meant through definitions and examples. Hopefully, the reason for doing this will become clear momentarily. Note also, I think these definitions will better coincide with what Mazure, Penrose and others who’ve written on this topic want.

Different Discovered worlds:
1. Physical world: Physical, 4 dimensional world. Meets criteria for Discovered.
2. Mental world: (ex: redness of an apple, the tone of a musical note, the sweetness of sugar, the sensation of making a choice) Not objectively measurable, so it doesn’t fit into the physical world. Meets criteria for Discovered.
3. Platonic Mathematical world: Per Penrose, Mazure, others. But is it really discovered?

These are the ONLY “Discovered” worlds. We might discover some unknown species of microbe on Mars for example, but that isn’t what is meant by discovered by Mazur and Penrose. For example, Mazur states:
If we adopt the Platonic view that mathematics is discovered, we are suddenly in surprising territory, for this is a full-fledged theistic position. Not that it necessarily posits a god, but rather that its stance is such that the only way one can adequately express one’s faith in it, the only way one can hope to persuade others of its truth, is by abandoning the arsenal of rationality, and relying on the resources of the prophets.

For the Platonists. One crucial consequence of the Platonic position is that it views mathematics as a project akin to physics, Platonic mathematicians being – as physicists certainly are describers or possibly predictors (I THINK HE’S REFERRING TO THE “PROPHETS” HERE) – not, of course, of the physical world, but of some other more noetic entity. Mathematics – from the Platonic perspective- aims, among other things, to come up with the most faithful description of that entity.
I’ll quote Penrose momentarily, but it seems obvious from the context that both Penrose and Mazur have something other in mind than simply the ‘discovery’ of life on Mars.

I think that most would agree these are different ‘worlds’ but that isn’t to indicate that they can exist independent from each other. For example, we might assume the mental world and the mathematical world are supervenient on the physical world. That is, the mental world requires the physical world to exist. The mathematical world might also be seen to require the physical world to exist. One might also argue that the mathematical world however, can’t exist without the mental world, so perhaps the mathematical world requires a mental world, which requires a physical world. Penrose would seem to suggest however, that each of the three above “worlds” are interrelated, and although they may require each other to exist, Penrose suggests these are to be seen as ‘sets’ analogous to mathematical sets, which overlap but have parts which DON’T OVERLAP! How can that be and how does he argue this?

I think first, we need to examine some examples of ‘inventions’ to understand what exists and how they relate to the above 3 potential ‘worlds’. Examples of inventions:
1. Things made of matter or energy: Exist in physical world. Sailboats, cars, monkeys, mountains, planets and galaxies are all made from matter/energy and exist in time and space. Thus, they are all inventions of the physical world since any specific one of them came about only because of happenstance.
2. Stories: Although a story can be written in a book, and the book exists in the physical world, the story itself can only have meaning if a mind is contemplating it. The actual story is invented and exists in the mental world.
3. Music: Again, there can be sound pressure waves which are part of the physical world, but the music itself, just like any qualia, exists only in the mental world. Music meets criteria for “invented”.
4. Art: Same as musical, but physically may include other forms of interactions such as a clay sculpture or light (em waves). Art is generally made of something physical but the appreciation of it as “art” is mental. Art is an invention.
5. The academic pursuit of physics, engineering, biology, etc…: These are all ‘ideas’ or models about the physical world which require a mental world and a mathematical description. Physical laws and various physical interactions are all modeled by these various areas of science. These models should be considered interpretations of the physical world, so all of these are inventions of the mental world as a minimum. Our interpretations are inventions, despite the fact that what we are working with is real and exists in the physical world.

Penrose argues for a “Platonic world of absolute mathematical forms” possessed by the physical world.
The very question of the internal consistency of a scientific model, in particular, is one that requires that the model be precisely specified. The required precision demands that the model be a mathematical one, for otherwise one cannot be sure that these questions have well-defined answers.
If the model itself is to be assigned any kind of ‘existence’, then this existence is located within the Platonic world of mathematical forms. Of course, one might take a contrary viewpoint: namely that the model is itself to have existence only within our various minds, rather than to take Plato’s world to be in any sense absolute and ‘real’. Yet there is something important to be gained in regarding mathematical structures as having a reality of their own.
(pg 12, The Road to Reality)

Section 1.4 (pg 17) begins his discussion of “three worlds and three deep mysteries”. His Figure 1.3 can be found on the web here: http://www.stefangeens.com/trinity.gif

In Figure 1.3, he shows what are sets. The Platonic mathematical world has some subset which contains or is projected upon the physical world. There is a subset of the physical world which is contains the mental world. And there is a subset of the mental world which contains the Platonic mathematical world. About this, he writes:
It may be noted, with regard to the first of these mysteries – relating the Platonic mathematical world to the physical world- that I am allowing that only a small part of the world of mathematics need have relevance to the workings of the physical world. It is certainly the case that the vast preponderance of the activities of pure mathematicians today has no obvious connection with physics, nor with any other science, although we may be frequently surprised by unexpected important applications. Likewise, in relation to the second mystery, whereby mentality comes about in association with certain physical structures (most specifically, healthy, wakeful human brains), I am not insisting that the majority of physical structures need induce mentality. While the brain of a cat may indeed evoke mental qualities, I am not requiring the same for a rock. Finally, for the third mystery, I regard it as self-evident that only a small fraction of our mental activity need be concerned with absolute mathematical truth! … These three facts are represented in the smallness of the base of the connection of each world with the next, the worlds being taken in a clockwise sense in the diagram.

Thus, according to Fig. 1.3, the entire physical world is depicted as being governed according to mathematical laws.
Penrose suggests that the mathematical world is discovered and is every bit as real as the mental world which is every bit as real as the physical world, albeit, real in a different sense of the term. He’s stating it is discovered because although nature obeys mathematical laws, there are ‘mathematical laws’ which have no application to the physical world, and these laws can only have a basis if there exists a mental world to contemplate them.

Anyway, that’s what Penrose seems to be saying. Here’s just one more from U of Oregon:
Thus, there came into existence two schools of thought. One that mathematical concepts are mere idealizations of our physical world. The world of absolutes, what is called the Platonic world, has existence only through the physical world. In this case, the mathematical world is the same as the Platonic world and would be thought of as emerging from the world of physical objects.

The other school is attributed to Plato, and finds that Nature is a structure that is precisely governed by timeless mathematical laws. According to Platonists we do not invent mathematical truths, we discover them. The Platonic world exists and physical world is a shadow of the truths in the Platonic world. This reasoning comes about when we realize (through thought and experimentation) how the behavior of Nature follows mathematics to an extremely high degree of accuracy. The deeper we probe the laws of Nature, the more the physical world disappears and becomes a world of pure math.

Mathematics transcends the physical reality that confronts our senses. The fact that mathematical theorems are discovered by several investigators indicates some objective element to mathematical systems. Since our brains have evolved to reflect the properties of the physical world, it is of no surprise that we discover mathematical relationships in Nature.

The laws of Nature are mathematical mostly because we define a relationship to be fundamental if it can be expressed mathematically.
Ref: http://abyss.uoregon.edu/~js/ast221/lectures/lec01.html
oldman
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#33
Feb16-08, 08:56 AM
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Quote Quote by Q_Goest View Post
Hi oldman,
So do you really live something like 100 km NW of Durban?
Since this thread peripherally involves different kinds of worlds --- the "real" world is one --- the answer to your question is: yes, I believe I do. Not at the end of the world; but you can see its edge from here.

Thanks very much for this long post. It's a humdinger, and I'll get back to you when I've read it carefully.
CaptainQuasar
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#34
Feb16-08, 06:00 PM
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Quote Quote by oldman View Post
(1) Herein lies a can of worms. The concept of a "field" (invented by Faraday?) is quintessentially abstract, and has proved endlessly useful in physics, right up to the idea of the inflaton field conjured out of thin air by cosmologists who promote the inflationary scenario. I must confess that I've used the concept myself, without much thought. But I'm pretty sure fields are not part of the "real" world -- just an very useful description of one of its aspects. So I'd say that patterns of fields are like mathematics -- invented rather than discovered.
But surely there are “real” things that behave like fields and interact with other “real” phenomena in a manner precisely analogous (isomorphic, I would say) to the mathematical construct of a field? Even if the field description of those real things is flawed or incomplete or an approximation. It seems to me no more invented than the consistent ratio between the diameter and circumference of circle-like objects.

Quote Quote by oldman View Post
(2) No, I haven't -- but thanks for the URL's. I've been aware of the structures created by this method, which I've classified (rather arbitrarily) as being the result of the clever "tricks" (“organizational attractors” to you!) devised by Conway for the lattice of cells he invented and that are played with. I would descibe this Game of Life as fooling around with structures that evolve --- but others in this forum would consider my use of the word "evolve" cavalier. And I've not read Wolram's book; only heard of it.
Now this particularly confuses me because one of the other things you identified as a “trick” was gravity! Do you not consider gravity to be real? It's not just a human invention, is it? (I think your critics for using “evolve” broadly might get a kick out of this…)
oldman
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#35
Feb17-08, 12:28 AM
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Quote Quote by CaptainQuasar View Post
But surely there are “real” things that behave like fields and interact with other “real” phenomena in a manner precisely analogous (isomorphic, I would say) to the mathematical construct of a field? Even if the field description of those real things is flawed or incomplete or an approximation. It seems to me no more invented than the consistent ratio between the diameter and circumference of circle-like objects.
I agree that there are "real things" that behave like fields. But you are missing the distinction between a description (field) and the thing being described (an interaction). Descriptions are always invented. Things ]may be discovered (Reflect on what we are --- nothing but a chattering species of African ape, driven to describe, and describe, and describe!).

In the past, a common way of describing physical interactions (like gravity) was to invoke the concept of "action at a distance". This is a not-very-clear invented description of interactions, which proved to be far less practical than the "field" concept which was invented to replace it. In fact it is no longer quite fashionable to talk simply of a "gravitational field" --- describing the metric "field" of spacetime with the help of the Riemann tensor seems more accurate.

Yet the mechanism/process by which mass/energy distorts spacetime is still quite unknown. It isn't with a pair of pliers. We don't even know if there is such a mechanism/process, and just accept the distortion as a given. This embarrassment is not quite swept under the carpet, but does seem to be ignored in polite general relativistic circles, perhaps because we are still too ignorant for this level to be usefully discussed.

But I agree that the classification of geometrical ratios, which you brought up earlier, may be a deeper question. Needs more thought.

...one of the other things you identified as a “trick” was gravity! Do you not consider gravity to be real? It's not just a human invention, is it?
No, of course it isn't. I don't understand gravity, but then I don't think anyone does, yet. What I meant was that gravity is a natural phenomenon, one description of which is:

"The 'trick' of nature that is ultimately responsible for the evolution (here I go again) of the physical universe --- from a (postulated) hot plasma into the galaxies, stars, planets and other debris we have discovered." Perhaps you can say something similar in a less clumsy way?
Hurkyl
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#36
Feb17-08, 12:45 AM
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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!


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