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The Question : is mathematics discovered or invented? 
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#19
Feb1508, 04:46 AM

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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!. 


#20
Feb1508, 04:54 AM

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#21
Feb1508, 09:04 AM

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http://www.physicsforums.com/showthread.php?t=215462 


#22
Feb1508, 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... 


#23
Feb1508, 05:24 PM

P: 705

From the other thread you linked to: ⚛



#24
Feb1508, 11:03 PM

P: 622

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 nonmystical 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. 


#25
Feb1508, 11:24 PM

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#26
Feb1608, 12:37 AM

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#27
Feb1608, 01:00 AM

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#28
Feb1608, 01:08 AM

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⚛



#29
Feb1608, 01:35 AM

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#30
Feb1608, 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 arrogantlynamed A New Kind Of Science. 


#31
Feb1608, 06:13 AM

P: 622

(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. 


#32
Feb1608, 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 antiPlatonist 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: 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. 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: Anyway, that’s what Penrose seems to be saying. Here’s just one more from U of Oregon: 


#33
Feb1608, 08:56 AM

P: 622

Thanks very much for this long post. It's a humdinger, and I'll get back to you when I've read it carefully. 


#34
Feb1608, 06:00 PM

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⚛



#35
Feb1708, 12:28 AM

P: 622

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 notveryclear 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. "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? 


#36
Feb1708, 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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