The Question : is mathematics discovered or invented?

[oldman]

(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]

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]

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]

(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.

(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…)

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[oldman]

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]

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!

 

[oldman]

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!

I agree. And we do have definitions: my Oxford dictionary defines invent as: Create by thought, originate, concoct. And Discover as: Find out that, become aware that. OK by me, and I'll try to use these definitions.

 

[oldman]

For me, your kind, detailed and thorough analysis of the positions taken by Mazur and Penrose (equivocation, leaning toward "Mathematics is discovered or become aware of"?) seems marred by your choice of definitions, which were:

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

But, to my delight this morning I found out that there are six new ducklings down at my dam. They're very much part of the natural world and weren't there yesterday. Does this mean that I didn't discover, or become aware of them? I suspect that they came into existence only because of a happenstance encounter of their mother with a drake. So must they have been invented? See the trouble one can have with such definitions!

Perhaps a little editing of your post in this respect would help me to answer it more coherently.

One remark: even eminent mathematicians like Penrose and Mazur are sometimes given to special pleading --- they love their subject so --- as we all love activities we excel at. It may be prudent to take their elevation of the nature of mathematics into an eternal truth with a pinch of salt.

 

[CaptainQuasar]

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!).

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. (Okay, we might have some fun talking about linguistics, but at least things like biology and chemistry and physics.)

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.

Come now - you aren't marshaling the argument I think you are, are you? Simply because something isn't understood that doesn't mean it's invented! In fact that seems oxymoronic - how could one not understand something that one has fully invented?

[EDIT] Re-reading I see that you did say that gravity isn't invented.

"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?

Alas, I might have some skill at saying things more specifically or more distinctively but they usually come out clumsier and less elegant.

―​

On the whole I think that it's a bit of a dodge (of the original question) to say that mathematics is invented in that it involves descriptions and other intermediate representations of the thing it is studying. (I'm speaking in general, not ascribing any malice to you.) 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.

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[JoeDawg]

A real problem I have with people referencing Plato's forms with regards to mathematics is that Plato believed his forms existed in 'another realm', a truer realm and that the universe was simply a shadow or manifestation of this truer reality. He wasn't talking about 'the mind' or some noumenal existence. (In my opinion, the elephant in this mathematical room is the 'mind of god') And yet we have no qualitative way of referring to this 'realm of math'. I think this is backwards thinking.

This is why I would say math is an invention. Math is a human description. Ideas abstracted from the observed. It is not some thing we found, full formed, on the side of the road.

Yes there are relationships present 'in the world', but that is not math.

The Pythagorean theorem works quite well, within a certain kind of space.
But even such concepts as 1 and 2, get fuzzy in relation to things like quantum entanglement. Its not that these concepts aren't useful in their proper place, nor that they exist in some nether realm, its that they are incomplete descriptions.

This is not to say that mathematics is not complex, and has no predictive value. But its predictive value is based on accepted axioms and in many cases is no more true than a scifi action novel. Its all about human imagination. My impression is that some pretentious mathematicians would find that sort of comparison embarrassing.

 

[Q_Goest]

Regarding the definitions, you won’t find Penrose nor Mazur try to define these terms. As Hurkyl mentions, and as yourself and CaptainQ continue to prove, it is the definitions that are eluding everyone and why it is so difficult to make sense of The Question. The question is more than just the use of a few words which propose a quandry. So when I say: “I think these definitions will better coincide with what Mazure, Penrose and others who’ve written on this topic want.” I also mean that I am assuming that the authors had something other in mind that what is normally conveyed by the conventional meaning of the terms “discover” and “invent”.

Clearly, we would all agree that it’s perfectly understandable when a newspaper talks about microbes being discovered on Mars or ducklings being discovered on a pond. I have no problem with those usages. However, Penrose puts out his figure and describes it without even using the terms discover or invent. Similarly, the U of Oregon doesn’t have those words in its description. Mazur thought he was being simple and clear by discovering that he could invent a way to use the terms in his paper, but here we are scratching our heads asking what these terms mean.

One way of getting rid of the problem is to try and define what the terms mean. Another way is to restate the question in a different way. What the authors want has to be taken in context.

 

[Q_Goest]

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. (Okay, we might have some fun talking about linguistics, but at least things like biology and chemistry and physics.)

I agree with oldman on this. The concept of gravity and electrical fields being some kind of three dimensional ‘thing’ is invented and is part of our human description of reality. I’d also say that all our descriptions of reality are invented, not discovered.

I’d agree with JoeDawg when he says:

The Pythagorean theorem works quite well, within a certain kind of space.
But even such concepts as 1 and 2, get fuzzy in relation to things like quantum entanglement. Its not that these concepts aren't useful in their proper place, nor that they exist in some nether realm, its that they are incomplete descriptions.

Penrose says something almost identical to this.

Similarly, all our scientific descriptions are invented because they are inexact.

 

[CaptainQuasar]

This is why I would say math is an invention. Math is a human description. Ideas abstracted from the observed. It is not some thing we found, full formed, on the side of the road.

Yes there are relationships present 'in the world', but that is not math.

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

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[CaptainQuasar]

I agree with oldman on this. The concept of gravity and electrical fields being some kind of three dimensional ‘thing’ is invented and is part of our human description of reality. I’d also say that all our descriptions of reality are invented, not discovered.

Of course descriptions are invented, they're things exclusively used by humans. That's practically a tautology. It's like saying that tools are invented or inventions are invented.

I really don't think any of this talk about descriptions being invented is at all addressing the question. It seems rather like eyebrow-arching sophistry to me.

The description of gravity is at least pointing to something more external to humanity than the description of Bilbo Baggins or the description of slapstick comedy is, right?

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[Q_Goest]

This is why I would say math is an invention. Math is a human description. Ideas abstracted from the observed.

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.

Clearly, math isn't only abstracted from what is observed in the physical world.

The Arabic number system is a base 10 language used to describe these seemingly superfluous mathematical concepts. Similarly, we can use a binary system or we could even use the Roman Numeral system which would make the math tremendously more cumbersome. I can't imagine trying to do my job as an engineer using the Roman numeral system, but I suppose it would be possible.

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.

 

[CaptainQuasar]

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.

That's simply like pointing out that “chicken” is a different word in Chinese than it is in English. That doesn't mean that chickens don't exist or that avian biology is a human invention.

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[Q_Goest]

The description of gravity is at least pointing to something more external to humanity than the description of Bilbo Baggins or the description of slapstick comedy is, right?

Yes, I agree there is a something real we are referring to when we talk about gravity. Could we describe it in some other way? I think we could. All I'm suggesting is that the descriptions are invented. I agree the physical interaction itself is part of the "physical world" that is being referred to by Penrose and Mazur.

If we're to distinguish between what is invented and what is discovered, we need to also make the distinction between our descriptions of reality and reality itself.

 

[Q_Goest]

That's simply like pointing out that “chicken” is a different word in Chinese than it is in English. That doesn't mean that chickens don't exist or that avian biology is a human invention.

Not sure what you're getting at. The Chinese person is referring to something and I'm referring to something. To extend that and say now that something is invented rather than discovered is a strawman agrument.

 

[CaptainQuasar]

Yes, I agree there is a something real we are referring to when we talk about gravity. Could we describe it in some other way? I think we could. All I'm suggesting is that the descriptions are invented. I agree the physical interaction itself is part of the "physical world" that is being referred to by Penrose and Mazur.

If we're to distinguish between what is invented and what is discovered, we need to also make the distinction between our descriptions of reality and reality itself.

Penrose and Mazur are starting to talk about the physical world, eh? Very avant-garde of them.

Yes, we could definitely describe things in a different way. This is partially what I mean by saying that numbers are more of a human construct and geometric ratios are more fundamental or more real and can probably be used to describe all the same things. It's like the way that a formal grammar and a finite automaton can be isomorphic or the way many topics in set theory and graph theory are isomorphic.

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[CaptainQuasar]

Not sure what you're getting at. The Chinese person is referring to something and I'm referring to something. To extend that and say now that something is invented rather than discovered is a strawman agrument.

You're not sure what I'm saying but it's a strawman?

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.

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[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.