from Stanford Encyclopedia of Philosophy
from wikipedia
Thursday, April 16, 2009
Monday, September 22, 2008
Plato
STILL DEBATING WITH PLATO
By Julie RehmeyerWeb edition : Friday, April 25th, 2008 Text Size Where do mathematical objects live?
Think too hard about it, and mathematics starts to seem like a mighty queer business. For example, are new mathematical truths discovered or invented? Seems like a simple enough question, but for millennia, it has provided fodder for arguments among mathematicians and philosophers.
Those who espouse discovery note that mathematical statements are true or false regardless of personal beliefs, suggesting that they have some external reality. But this leads to some odd notions. Where, exactly, do these mathematical truths exist? Can a mathematical truth really exist before anyone has ever imagined it?
On the other hand, if math is invented, then why can’t a mathematician legitimately invent that 2 + 2 = 5?
Many mathematicians simply set nettlesome questions like these aside and get back to the more pleasant business of proving theorems. But still, the questions niggle and nag, and every so often, they rise to attention. Several mathematicians will ponder the question of whether math is invented or discovered in the June European Mathematical Society Newsletter.
Plato is the standard-bearer for the believers in discovery. The Platonic notion is that mathematics is the imperturbable structure that underlies the very architecture of the universe. By following the internal logic of mathematics, a mathematician discovers timeless truths independent of human observation and free of the transient nature of physical reality. “The abstract realm in which a mathematician works is by dint of prolonged intimacy more concrete to him than the chair he happens to sit on,” says Ulf Persson of Chalmers University of Technology in Sweden, a self-described Platonist.
The Platonic perspective fits well with an aspect of the experience of doing mathematics, says Barry Mazur, a mathematician at Harvard University, though he doesn’t go so far as to describe himself as a Platonist. The sensation of working on a theorem, he says, can be like being “a hunter and gatherer of mathematical concepts.”
But where are those hunting grounds? If the mathematical ideas are out there, waiting to be found, then somehow a purely abstract notion has to have existence even when no human being has ever conceived of it. Because of this, Mazur describes the Platonic view as “a full-fledged theistic position.” It doesn’t require a God in any traditional sense, but it does require “structures of pure idea and pure being,” he says. Defending such a position requires “abandoning the arsenal of rationality and relying on the resources of the prophets.”
Indeed, Brian Davies, a mathematician at King's College London, writes that Platonism “has more in common with mystical religions than with modern science.” And modern science, he believes, provides evidence to show that the Platonic view is just plain wrong. He titled his article “Let Platonism Die.”
If mathematics is the perception of this realm of pure ideas, then doing mathematics requires our brains to somehow reach beyond the physical world. Davies argues that brain-imaging studies are making this belief steadily less plausible. He points out that our brains integrate many different aspects of visual perception with memory and preconceptions to create a single image — not always correctly, as optical illusions make clear. He also says that brain-imaging studies are beginning to show the biological basis of our numeric sense.
But Reuben Hersh of the University of New Mexico isn’t convinced that studies like these logically destroy the Platonic notion of an intuitive faculty to perceive mathematics. Nevertheless, he rejects the Platonic view, arguing instead that mathematics is a product of human culture, not fundamentally different from other human creations like music or law or money.
The challenge, he admits, is to explain why it is that mathematical statements can be definitively true or false, not subject to taste or whim. With simple statements like “2 + 2 = 4,” this is because of the connection between mathematics and physics, he says. Such a statement describes, for example, the way that coins or buttons behave. For more abstract statements that are further removed from the physical world, he points to the structure of our brains and our penchant for logic.
But Mazur finds that explanation unsatisfying. “We should keep an eye on the stealth word ‘our,’” he writes. “Is the we meant to be each and every one of us, given our separate and perhaps differing and often faulty faculties?” In this case, mathematics itself has to vary as individuals do.
On the other hand, if “we” means a kind of abstraction of our individual capabilities — the common thing that binds us together without actually being any of us — he says that we are verging back toward the Platonic notion of a realm of abstract ideas.
But the notion of invention also captures something true about the experience of doing mathematics, in his view. “At times,” he says, “I seem to be engaged in an analysis of my thought processes or other people’s thought processes while doing mathematics.” All aspects of these experiences, he argues, need to be included in these discussions.
“One thing is — I believe — incontestable,” he writes. “If you engage in mathematics long enough, you bump into The Question, and it won’t just go away. If we wish to pay homage to the passionate felt experience that makes it so wonderful to think mathematics, we had better pay attention to it.”
References:
Hersh, R. 2008. On Platonism. European Mathematical Society Newsletter (June). http://www.ems-ph.org/journals/journal.php?jrn=news.
Mazur, B. 2008. Mathematical Platonism and its Opposites. European Mathematical Society Newsletter (June). http://www.ems-ph.org/journals/journal.php?jrn=news.
Persson, U. 2008. On Platonism. European Mathematical Society Newsletter (June). http://www.math.chalmers.se/Platonism/platonism.pdf.
Davies, E.B. 2007. Let Platonism Die. European Mathematical Society Newsletter (June). http://www.ems-ph.org/journals/newsletter/pdf/2007-06-64.pdf.
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Comments 5
Does the answer have to be one or the other? Don't we invent the postulates, keep the ones that seem to have relevance to our world and discard the ones that don't, then discover the consequences of accepting those postulates? rocketlady Jun. 15, 2008 at 2:45pm PB FTW!!! LOOPY Apr. 28, 2008 at 7:01pm I submit that we resist the urge to over-think this idea.
Mathematics is a language that helps communicate the physicalness of our universe & beyond.
What is, is. What was, was. What will be, will be.
Math allows us the luxury of seeing things that our eyes cannot. Our discoveries do not include the materialization of our find; we found a way to see them mathematically as they exist.
Let us leave the mysteries of THE QUESTION for another day when we have a good bottle of tequila & an afternoon to dream. Ralph's Alter Ego Apr. 28, 2008 at 4:58pm I thought that Gödel numbers, the tool he created to derive his incompeteness theorem allowed to map formal systems into natural nubers, and to derive proofs using only simple arithmetics.
This would nuke the argument that the more abstract statements don't relate to the physical world (at least for formal statements in classical mathematics and computer science (see Curry-Howard isomorphism)).
I use the conditional here because this is not my primary field, although I love this kind of arguments...
I don't buy the neuroscience-based argument either (more of my field here). Our ability to represent something in our mind doesn't prevent it from existing outside of it.
I don't choke at the idea that mathematics (at least in axiomatic form) are embedded in the very fabric of the universe (whose existence is in my eye an intractable mystery). The very mathematical nature of physics pleads for this.
Another interresting point is that our brains are part of the universe. Assuming matter/energy and the laws governing them as "axioms", the "invention" of a theorem would also be the discovery of it's neuronal hence physical representation.
I hope this doesn't sound too cranky :-). Flirpon Barzadrouche Apr. 28, 2008 at 5:33am The whole point of mathematics being discovered instead of invented is
that there must have been an original designer of those equations, and that all things that exist work in accordance to them. Historically, this is something that scientists and mathematicians have been discovering since the age of enlightenment and before.
Naturally, a lot of us are on the side of mathematics being discovered rather than invented. Perhaps this is because it is written in Scripture that God has set eternity in the hearts of man, and therefore, the math he has invented and used to build his creation are also discoverable by this creation of his who was made in his image.
Many people can see the absolute beauty that exists in his mathematics and in the design of everything that conforms to those equations. Even in DNA, the math is both amazing and far more intricate than we have yet understood. The Scriptures also say that the evidence of God's existence is to be seen in all creation, and that "in him we live, and move and have our being." So, yes, there are real and vital implications to believing that mathematics is discovered instead of invented.
Are you sure you want to go there with THE QUESTION? Roger Born Apr. 27, 2008 at 9:02am
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By Julie RehmeyerWeb edition : Friday, April 25th, 2008 Text Size Where do mathematical objects live?
Think too hard about it, and mathematics starts to seem like a mighty queer business. For example, are new mathematical truths discovered or invented? Seems like a simple enough question, but for millennia, it has provided fodder for arguments among mathematicians and philosophers.
Those who espouse discovery note that mathematical statements are true or false regardless of personal beliefs, suggesting that they have some external reality. But this leads to some odd notions. Where, exactly, do these mathematical truths exist? Can a mathematical truth really exist before anyone has ever imagined it?
On the other hand, if math is invented, then why can’t a mathematician legitimately invent that 2 + 2 = 5?
Many mathematicians simply set nettlesome questions like these aside and get back to the more pleasant business of proving theorems. But still, the questions niggle and nag, and every so often, they rise to attention. Several mathematicians will ponder the question of whether math is invented or discovered in the June European Mathematical Society Newsletter.
Plato is the standard-bearer for the believers in discovery. The Platonic notion is that mathematics is the imperturbable structure that underlies the very architecture of the universe. By following the internal logic of mathematics, a mathematician discovers timeless truths independent of human observation and free of the transient nature of physical reality. “The abstract realm in which a mathematician works is by dint of prolonged intimacy more concrete to him than the chair he happens to sit on,” says Ulf Persson of Chalmers University of Technology in Sweden, a self-described Platonist.
The Platonic perspective fits well with an aspect of the experience of doing mathematics, says Barry Mazur, a mathematician at Harvard University, though he doesn’t go so far as to describe himself as a Platonist. The sensation of working on a theorem, he says, can be like being “a hunter and gatherer of mathematical concepts.”
But where are those hunting grounds? If the mathematical ideas are out there, waiting to be found, then somehow a purely abstract notion has to have existence even when no human being has ever conceived of it. Because of this, Mazur describes the Platonic view as “a full-fledged theistic position.” It doesn’t require a God in any traditional sense, but it does require “structures of pure idea and pure being,” he says. Defending such a position requires “abandoning the arsenal of rationality and relying on the resources of the prophets.”
Indeed, Brian Davies, a mathematician at King's College London, writes that Platonism “has more in common with mystical religions than with modern science.” And modern science, he believes, provides evidence to show that the Platonic view is just plain wrong. He titled his article “Let Platonism Die.”
If mathematics is the perception of this realm of pure ideas, then doing mathematics requires our brains to somehow reach beyond the physical world. Davies argues that brain-imaging studies are making this belief steadily less plausible. He points out that our brains integrate many different aspects of visual perception with memory and preconceptions to create a single image — not always correctly, as optical illusions make clear. He also says that brain-imaging studies are beginning to show the biological basis of our numeric sense.
But Reuben Hersh of the University of New Mexico isn’t convinced that studies like these logically destroy the Platonic notion of an intuitive faculty to perceive mathematics. Nevertheless, he rejects the Platonic view, arguing instead that mathematics is a product of human culture, not fundamentally different from other human creations like music or law or money.
The challenge, he admits, is to explain why it is that mathematical statements can be definitively true or false, not subject to taste or whim. With simple statements like “2 + 2 = 4,” this is because of the connection between mathematics and physics, he says. Such a statement describes, for example, the way that coins or buttons behave. For more abstract statements that are further removed from the physical world, he points to the structure of our brains and our penchant for logic.
But Mazur finds that explanation unsatisfying. “We should keep an eye on the stealth word ‘our,’” he writes. “Is the we meant to be each and every one of us, given our separate and perhaps differing and often faulty faculties?” In this case, mathematics itself has to vary as individuals do.
On the other hand, if “we” means a kind of abstraction of our individual capabilities — the common thing that binds us together without actually being any of us — he says that we are verging back toward the Platonic notion of a realm of abstract ideas.
But the notion of invention also captures something true about the experience of doing mathematics, in his view. “At times,” he says, “I seem to be engaged in an analysis of my thought processes or other people’s thought processes while doing mathematics.” All aspects of these experiences, he argues, need to be included in these discussions.
“One thing is — I believe — incontestable,” he writes. “If you engage in mathematics long enough, you bump into The Question, and it won’t just go away. If we wish to pay homage to the passionate felt experience that makes it so wonderful to think mathematics, we had better pay attention to it.”
References:
Hersh, R. 2008. On Platonism. European Mathematical Society Newsletter (June). http://www.ems-ph.org/journals/journal.php?jrn=news.
Mazur, B. 2008. Mathematical Platonism and its Opposites. European Mathematical Society Newsletter (June). http://www.ems-ph.org/journals/journal.php?jrn=news.
Persson, U. 2008. On Platonism. European Mathematical Society Newsletter (June). http://www.math.chalmers.se/Platonism/platonism.pdf.
Davies, E.B. 2007. Let Platonism Die. European Mathematical Society Newsletter (June). http://www.ems-ph.org/journals/newsletter/pdf/2007-06-64.pdf.
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Found in: Numbers
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Comments 5
Does the answer have to be one or the other? Don't we invent the postulates, keep the ones that seem to have relevance to our world and discard the ones that don't, then discover the consequences of accepting those postulates? rocketlady Jun. 15, 2008 at 2:45pm PB FTW!!! LOOPY Apr. 28, 2008 at 7:01pm I submit that we resist the urge to over-think this idea.
Mathematics is a language that helps communicate the physicalness of our universe & beyond.
What is, is. What was, was. What will be, will be.
Math allows us the luxury of seeing things that our eyes cannot. Our discoveries do not include the materialization of our find; we found a way to see them mathematically as they exist.
Let us leave the mysteries of THE QUESTION for another day when we have a good bottle of tequila & an afternoon to dream. Ralph's Alter Ego Apr. 28, 2008 at 4:58pm I thought that Gödel numbers, the tool he created to derive his incompeteness theorem allowed to map formal systems into natural nubers, and to derive proofs using only simple arithmetics.
This would nuke the argument that the more abstract statements don't relate to the physical world (at least for formal statements in classical mathematics and computer science (see Curry-Howard isomorphism)).
I use the conditional here because this is not my primary field, although I love this kind of arguments...
I don't buy the neuroscience-based argument either (more of my field here). Our ability to represent something in our mind doesn't prevent it from existing outside of it.
I don't choke at the idea that mathematics (at least in axiomatic form) are embedded in the very fabric of the universe (whose existence is in my eye an intractable mystery). The very mathematical nature of physics pleads for this.
Another interresting point is that our brains are part of the universe. Assuming matter/energy and the laws governing them as "axioms", the "invention" of a theorem would also be the discovery of it's neuronal hence physical representation.
I hope this doesn't sound too cranky :-). Flirpon Barzadrouche Apr. 28, 2008 at 5:33am The whole point of mathematics being discovered instead of invented is
that there must have been an original designer of those equations, and that all things that exist work in accordance to them. Historically, this is something that scientists and mathematicians have been discovering since the age of enlightenment and before.
Naturally, a lot of us are on the side of mathematics being discovered rather than invented. Perhaps this is because it is written in Scripture that God has set eternity in the hearts of man, and therefore, the math he has invented and used to build his creation are also discoverable by this creation of his who was made in his image.
Many people can see the absolute beauty that exists in his mathematics and in the design of everything that conforms to those equations. Even in DNA, the math is both amazing and far more intricate than we have yet understood. The Scriptures also say that the evidence of God's existence is to be seen in all creation, and that "in him we live, and move and have our being." So, yes, there are real and vital implications to believing that mathematics is discovered instead of invented.
Are you sure you want to go there with THE QUESTION? Roger Born Apr. 27, 2008 at 9:02am
post a comment
Please login or register to participate.
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Thursday, April 24, 2008
Confessions of a Chinese Sudoku Addict
Playing evil level sudoku on websodku.com without marks.. best time reaches something above 7 minutes.. getting closer to a sudoku expert, and hence closer to get rid of the addiction.
*** some days ago ***
A discussion on mathematical logic becomes exchanging ideas about solving sudokus. Unfortunately, I don't know much about it, even though I am a self-claimed "sudoku addict". Yes, there are gamblers who don't really know how to gamble correctly --- by which we mean knowing the winning strategy, as if that existed --- and as addicted to sudoku as I am, solving a hard sudoku without using marks may take for ever, or that I just give it up. I do suspect it is being a bad sudoku solver that makes me addicted to this game, or at least there are some connections between these two facts.
Rumor has it that there are relations between sudoku and mathematics. Believing it was true, it sounded to us a good idea to use this topic in our summer workshop, more explicitly, to talk about sudoku solving as a means to teach mathematical logic. I love this game, and at the same time, I doubt how much amount of "real math" is contained in it. Yes I have a math friend who plays sudoku very well, solving hardest ones on unisudoku within 2 and a half minutes. There is even this latex package called sudoku, making it easy to draw sudoku diagrams without using messy environments such as array or tabular. It is undoubtful that sudoku makes a good topic in blogging or math education or daily chat...
OK, come back to the discussion. It is interesting since it let me start to summarize my sudoku play, which is good, especially at this point when I am desperately addicted and really bad at it. As soon as I claimed that I was a sudoku addict, my boss asked me in which way I played, by which he meant using or not using marks. I did not admit that I could not play without marks, so I said it depended, blahblah.. As a matter of fact, even when using marks my best time ever on a hard sudoku was something more than 7 minutes, with a lot of luck and the use of a mouse. Without marks my average time would be above 20 minutes for sure... We then talked about sudoku strategies, catogorized by my boss into two general sorts: finding-the-missing-part sort and eliminating sort. I regard the first sort much harder than the latter. We also talked about systems we used in playing sudokus. My system is to go through 1 to 9 and to come back to 1 again. His system is a random one, finding whatever block that can be filled with some number. It is my opinion that to play randomly might be relatively more advantageous to non-mark players, while going in order is important for marking, that is, for not missing any possible marks.
There are a lot of papers on sudoku now, which I generally found it boring to read. Some while ago when I complained to my friend what a bad player I was, he asked me how long I had been playing. Well, for me it was a couple of days back then, but he had played for years. A couple of weeks have passed since then, and I haven't improved into any comfortable level. Sooner or later, I'd have to make a choice between two things: one is to keep being addicted for a couple of years and become a sudoku expert, the other is to forget about this game and set myself free. It is a hard one.
*** some days ago ***
A discussion on mathematical logic becomes exchanging ideas about solving sudokus. Unfortunately, I don't know much about it, even though I am a self-claimed "sudoku addict". Yes, there are gamblers who don't really know how to gamble correctly --- by which we mean knowing the winning strategy, as if that existed --- and as addicted to sudoku as I am, solving a hard sudoku without using marks may take for ever, or that I just give it up. I do suspect it is being a bad sudoku solver that makes me addicted to this game, or at least there are some connections between these two facts.
Rumor has it that there are relations between sudoku and mathematics. Believing it was true, it sounded to us a good idea to use this topic in our summer workshop, more explicitly, to talk about sudoku solving as a means to teach mathematical logic. I love this game, and at the same time, I doubt how much amount of "real math" is contained in it. Yes I have a math friend who plays sudoku very well, solving hardest ones on unisudoku within 2 and a half minutes. There is even this latex package called sudoku, making it easy to draw sudoku diagrams without using messy environments such as array or tabular. It is undoubtful that sudoku makes a good topic in blogging or math education or daily chat...
OK, come back to the discussion. It is interesting since it let me start to summarize my sudoku play, which is good, especially at this point when I am desperately addicted and really bad at it. As soon as I claimed that I was a sudoku addict, my boss asked me in which way I played, by which he meant using or not using marks. I did not admit that I could not play without marks, so I said it depended, blahblah.. As a matter of fact, even when using marks my best time ever on a hard sudoku was something more than 7 minutes, with a lot of luck and the use of a mouse. Without marks my average time would be above 20 minutes for sure... We then talked about sudoku strategies, catogorized by my boss into two general sorts: finding-the-missing-part sort and eliminating sort. I regard the first sort much harder than the latter. We also talked about systems we used in playing sudokus. My system is to go through 1 to 9 and to come back to 1 again. His system is a random one, finding whatever block that can be filled with some number. It is my opinion that to play randomly might be relatively more advantageous to non-mark players, while going in order is important for marking, that is, for not missing any possible marks.
There are a lot of papers on sudoku now, which I generally found it boring to read. Some while ago when I complained to my friend what a bad player I was, he asked me how long I had been playing. Well, for me it was a couple of days back then, but he had played for years. A couple of weeks have passed since then, and I haven't improved into any comfortable level. Sooner or later, I'd have to make a choice between two things: one is to keep being addicted for a couple of years and become a sudoku expert, the other is to forget about this game and set myself free. It is a hard one.
Tuesday, April 22, 2008
Supreme Fascist
from a page on everything:
As Paul Hoffman's excellent book on Erdös, The Man who loved only numbers describes, The SF is the Supreme Fascist, the Number-One Guy Up There, God, who was always tormenting Erdös by hiding his glasses, stealing his Hungarian passport, or worse yet, keeping to himself the elegant solutions to all sorts of intriguing mathematical problems.
Another piece of Erdösese (other examples include "epsilon" for small child, "bosses" for women and "slaves" for men), his view of God as the Supreme Fascist is particularly pessimistic:
"The game of life, is to keep the SF's score low. If you do something bad in life, the SF gets two points. If you don't do something good that you should have done, the SF gets one point. You never score, so the SF always wins." - Erdös.
Erdös first began calling God the SF in the 1940s. "With so many bad things in the world, I'm not sure that God, should He exist, is good". He voiced approval for a novel, The Revolt of the Angels by Anatole France, which depicted God as evil and the Devil as benign.
Of particular note is the SF's book, transfinite in size, which "contains the best proofs of all mathematical theorems, proofs that are elegant and perfect". Thus the greatest praise one could receive from Erdös would be for him to describe your work as "straight from the book". He also observed that "You don't have to believe in God, but you should believe in the Book", illustrating his belief that the best mathematics wasn't just functional but beautiful.
As Paul Hoffman's excellent book on Erdös, The Man who loved only numbers describes, The SF is the Supreme Fascist, the Number-One Guy Up There, God, who was always tormenting Erdös by hiding his glasses, stealing his Hungarian passport, or worse yet, keeping to himself the elegant solutions to all sorts of intriguing mathematical problems.
Another piece of Erdösese (other examples include "epsilon" for small child, "bosses" for women and "slaves" for men), his view of God as the Supreme Fascist is particularly pessimistic:
"The game of life, is to keep the SF's score low. If you do something bad in life, the SF gets two points. If you don't do something good that you should have done, the SF gets one point. You never score, so the SF always wins." - Erdös.
Erdös first began calling God the SF in the 1940s. "With so many bad things in the world, I'm not sure that God, should He exist, is good". He voiced approval for a novel, The Revolt of the Angels by Anatole France, which depicted God as evil and the Devil as benign.
Of particular note is the SF's book, transfinite in size, which "contains the best proofs of all mathematical theorems, proofs that are elegant and perfect". Thus the greatest praise one could receive from Erdös would be for him to describe your work as "straight from the book". He also observed that "You don't have to believe in God, but you should believe in the Book", illustrating his belief that the best mathematics wasn't just functional but beautiful.
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