About a year and a half ago, after reading Penrose's book "The Road to Reality" I started a thread The Platonic World. Real or a human construct? in which I tapped into a rich vein of opinion concerning the nature of mathematics. At the time I hadn't appreciated how strongly some mathematicians feel about this, so I trod on some toes, and also got diverted into posting about string theory. I've recently been enlightened by a new article that may be of interest in this forum. It's perhaps better to start a new thread about it than to revive the old one. The new article is Mathematical Platonism and its Opposites , by Barry Mazur. He holds a name chair at Harvard and the article is in my opinion worth reading, perhaps by philosophers especially. He discusses both Platonic and Anti-Platonic views about this question, which he says concern most dedicated mathematicians. Do folk in this forum have any further comments on what Mazur calls "The Question" that they might wish to express ?
A vote for both To invent means to originate as a product of one's own ingenuity, experimentaion, or contrivance. Mathematics and its subjects are abstract ideas. Abstractions must be ingeniously created using mental processes includiing negation. analogizing, and word creation. To discover means to gain sight or knowledge of (something previously unseen or unknown). When an abstraction is created, its creator gains knowledge of something previously unknown. Therefore, mathematics is both invented and discovered.
Hi oldman. I read the part in Penrose's book you're refering to and agree it's a "rich vein of opinion concerning the nature of mathematics". I enjoy Penrose. Your link to the paper doesn't work. The link somehow doubled up on the http:// part... here's the link: http://www.math.harvard.edu/~mazur/papers/plato4.pdf I read it through and have a few thoughts. I'll have to come back to this. Just wanted to fix the link. You might want to edit the OP and fix the link. Try "Preview Post" and verify it works prior to "Submit Reply".
I hate to sound repetitive but- mathematics is both invented and discovered. It is invented when we create the axioms for a new mathematical system, discovered when we determine what theorems are true in that system. It's hard to get a good argument going when everyone agrees!
Thanks for this help. I've replaced the link, and it seems to work now. Let me know if there's still a problem.
Well, not quite everyone. This statement ("mathematics is both invented and discovered") is sophistry, and poor sophistry at that. When axioms are created, the rest of the mathematical system doesn't spring into existence magically fully formed, like Eve from Adam's rib; it has to be deduced as an abstract logical structure, which is an invented extension of the invented axioms. You can't have your cake and eat it, as it were! Yes, mathematics is indeed nothing more than a (very complex) language, Morodin, but languages evolve as invented constructs. They aren't discovered.
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?
If there are two trees growing in solitude on a hill top, is there any way to conclude without consciousness that there are two trees there? It is based upon perception, and then a brain processes the information into what makes sense to it. You could in some ways say that there are two trees independently of any mind, brain or perception, but that's treading on a fine line between objectivity and subjectivity. I would rather just say that "the world is what it is" and then leave my brain to do the categorizing. Same with math then, as ivy said, we create something, and then see if it fits, of which is dependent on the actual reality of the situation, the math itself is an abstracted view on reality, it is just one of probably many ways of perceiving the world through a language.
Math is an invented symbol system... or language. What is discovered, is whether mathematical representations agree with 'reality'. When we find they do or do not agree, we build on that. We invent new ways to use the symbols and then... once again... compare it to reality..
Yes, I agree with (1). I'm not so sure about (2). Does one "discover" that your fingers are five in number, having first invented the concept "five", or do you simply discover that this invented description helps you to talk about your fingers and be understood? The Question that Barry Mazur wrote about is not trivial, and my experience so far is that most people hedge about answering it, because they are not sure of being correct. This includes Penrose, another mathematician of considerable distinction. As you will know (having read his book) Penrose suggested that there are three worlds to consider --- Platonic Mathematical, Mental and Physical. He argued that the links between them are mysterious. But I don't think he provided any hard evidence for their separate existence. I think that Penrose would opt for "discovered". But he doesn't explicitly plump for this choice, as far as I can see. Although he elaborates very persuasively on this theme, I am still sceptical of his thesis. His discussions set me thinking and reading; for example Susan Haack's "Defending Science". Now there's a philosopher with her feet on the ground. I like the way Mazur phrased The Question; it's simple and direct, and has evoked some straightforward opinions here. Most ordinary folk (especially myself) find the high peaks (and sometimes the foothills) of mathematics a steep climb, often too steep. Mathematical experts, on the other hand, are so engaged in the technicalities of scrambling around among these peaks that they find it difficult to consider without prejudice the simple nature of mathematical elements. Even in Mazur's case his prejudice -- which is a delight in mathematics - seeps through. That's why I've questioned the nature of simple stuff like arithmetic and counting numbers rather than the fabric of the wonderful tapestry of modern mathematics.
oldman, My point about consciousness can be understood by your question; All I meant was that it's impossible to come to a conclusion about this without first acknowledging that you're making a conscious effort to do so. That's why I said 'the world is what it is' because there exists a hand, with fingers on it, and the hand and the fingers are abstractions, arguing whether they exist separately or what quantity they come in is something a brain has to do. Objectively the world will never be anything more than just 'what it is.' I would also say that there is no real answer to this question, because you're asking us to make an objective claim regarding quantity which is not possible from a subjective point of view.. It would all in the end be something the brain decides.
Well, I'm inclined to try staking out the position that it's [highlight]entirely discovered[/highlight]. We've got a language for it and names and symbols for things, but the same is true of biology and we didn't invent the species and subjects of study there. (At least, not until the science of genetic engineering showed up and started working on my army of flying monkeys.) I think it's easiest to see in geometry. There are many spherical bodies in the universe. A plane cut through any of them exposes a circle and the ratio of the diameter of that circle to its circumference approaches π, whether or not there are humans there to see it. I think that all of mathematics must be like this: “real” congruities in the universe itself that the faculty of reason allows us to perceive. We may only be perceiving a small, warped part of it: for example, numbers may not actually be as important as they appear to us to be, perhaps it is all better expressed and comprehended as geometric ratios rather than the abstract symbols that are easiest for our minds to handle. But mathematics is external to humans in its entirety, written out in the Book of the Cosmos, in no way a human creation.⚛
Ask a very young child how many fingers they have. They learn number systems that were invented. Its fairly meaningless to them, until they can apply it to their fingers. A number system that doesn't describe reality is discarded. Its wrong to say 1+1=3, because that doesn't describe reality. As children we learn early the 'object' model of our universe, because it is useful at the 'level' we exist, as opposed to say, the quantum, or cosmic levels. We learn that things seem distinct from each other and that some of those things are similar. We group those things and ascribe value to them. We invent, or learn what others have invented, a way of abstracting objects and referring to them with numbers. This object model is really just a level of abstraction created by our limited senses.
Thanks, o'pod, I don't disagree with you: I agree that we have no option but to accept the world for "what it is". I would only add that I fear we do so with little prospect of fully understanding why we are presented with all this stuff, and what we ourselves are. But I hope that in the meantime, as a practical matter, there is some prospect of categorising our everyday physical world as "real" and mathematics as "abstract" without worrying too much about subjectivity and objectivity, and then deciding whether the latter was discovered or invented by us. I'm asking only for a simply substantiated claim, not necessarily an objective one.
Belay this! Cap'n. Biology doesn't fit into the same box as mathematics. In biology is described, with an invented language if need be, the details and workings of critters like your flying monkeys, which are part of the discovered physical world. Mathematics, on the other hand, is (I maintain) itself a language that is used to describe other discovered aspects of the physical world. Geometry treats spatial relationships between shapes and finds (as you describe) that pi is a number. By the way, its value depends on what kind of geometry you use. It ain't necessarily near 3.14159 unless the geometry of the discovered shape that humans (who must be there to invent the describing, measuring and arithmetic of division) decide is Euclidean. This is a bold statement that you need to amplify; especially about the Book of the Cosmos. I'll buy it if you tell me where to do so! P.S. Every morning I get gabbled at by my wild Vervets, who fortunately can't fly.
This actually kind of makes my point about geometric ratios being more fundamental than numbers. The thing is that there's a consistent ratio between the equivalents of the diameter and circumference within the equivalent of a circle in any given geometry, circles which are represented in the world outside of humans. Whether that ratio is “really” π or not as represented in these symbols we call numbers. Whatever that ratio is, in whatever geometry system you might view it through, it's going to be equally related to the sine, cosine, and tangent function-equivalents of the equivalents of triangles, and thereby related to the equivalent of wave mechanics and the manifold phenomena we have seen to be governed by wave mechanics. However anthropocentric mathematics is there's something “real” that is isomorphic to it and embedded in the universe. (I'm familiar with spherical geometry and hyperbolic geometry and others but I didn't go and dig out my old textbooks and figure out whether what I've said above is true, so feel free to shred it to pieces if it isn't. :tongue2:) Just a flowery metaphor for the underlying reality of the cosmos, borrowed from Omar Khayyám:The Moving Finger writes; and, having writ, Moves on: nor all your Piety nor Wit Shall lure it back to cancel half a Line, Nor all your Tears wash out a Word of it⚛
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!.
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.