cpshelley@violet.uwaterloo.ca (cameron shelley) (10/05/90)
In article <3549@media-lab.MEDIA.MIT.EDU> minsky@media-lab.media.mit.edu (Marvin Minsky) writes: > < (Point 2) I sometimes wonder how much our theories are not just >recastings of our experience (without great insight). It has happened >several times that physicists have found in the mathematics literature >exactly the math they need to solve their physics problems. Whence >came the mathematics? Was it not from abstractions of earlier physics >problems (an historian of science should be able to prove or disprove >this conjecture)? > While I'm not a historian of science, I believe that the first *recorded* use of mathematics was in account keeping, using cuneiform notation to keep track of debts/payments. The Babylonians developed alot of math in the name of astrology and numerology - which are both widespread to this day. They were considered then as we consider science now. Thus rather than being abstractions of something as dry as physics problems, they might be considered abstractions of personal problems - from the individual to the political level. For another example, look at how the pythagoreans regarded their mathematical achievements. >Indeed, I have heard science historians argue that much of mathematics >came from earlier physics theories. Doubtful, I think. The ancients regarded mathematics as mostly a mental discipline which was learned through initiation into whatever school the well-to-do man could find palatable. One student at Plato's academy (whose name I don't recall) was tossed out after asking "what good geometry is"! There were, as always, exceptions to this rule. Btw, I think many people still regard mathematics this way today... :> > But there is another possibility >explained in an essay of mine --- "Communication with Alien >Intelligence," in @i[Extraterrestrial: Science and Alien >Intelligence,] (E. Regis, ed.) Cambridge University Press, 1985. This >is a cute theory based on some experiments with very small Turing >machines. It turned out that many of them performed operations that >could be interpreted as elementary addition -- while none of them did >anything that was "similar" to addition but not exactly addition! > >What that seems to mean is that the most elementary mathematics -- or, >rather, the kinds that humans have historically first imagined -- hold >a peculiar position among "all possible mathematical systems". In a >sense, they might simply be the ones that are "easiest for a machine >to think of". Do you mean "easiest for a human to think of"? If they're so simple, why has it taken us so long to develop them? It wasn't until the renaissance (sp?) that anyone really thought of modelling the world in mathematical terms alone. People must be indoctrinated for years before the conventions of math and physics seem 'natural'. > Why, then, might they help in making physics theories? >Either because the universe, too, is peculiarly simple -- whatever >that means -- or that the simplest theories are (at least ,at first) >the most useful ones -- simply because they are the first ones we can >use to make any predictions at all... This sounds somewhat too idealized to me. The development of any kind of theories historically have been largely done by adapting old ideas and conventions, even where they are out of context (analogy). This implies more of a feedback mechanism of conception on conception than a progression of theories corrected by regular reference to "the universe" as you seem to be saying. The former would be much more haphazard and less 'logical' than than the second (and take much more time and arguement), which seems to agree better with the actual history of the sciences. :> In other words, the way we look at the world is perhaps more arbitrary than you think - although I cannot produce any evidence to support this, not having a sufficiently non-human perspective to refer to! -- Cameron Shelley | "Armor, n. The kind of clothing worn by a man cpshelley@violet.waterloo.edu| whose tailor is a blacksmith." Davis Centre Rm 2136 | Phone (519) 885-1211 x3390 | Ambrose Bierce
jjewett@math.lsa.umich.edu (Jim Jewett) (10/08/90)
In article <1990Oct4.175806.7711@watdragon.waterloo.edu>, cpshelley@violet.uwaterloo.ca (cameron shelley) writes: |> In article <3549@media-lab.MEDIA.MIT.EDU> minsky@media-lab.media.mit.edu (Marvin Minsky) writes: |> > But there is another possibility |> >explained in an essay of mine --- "Communication with Alien |> >Intelligence," in @i[Extraterrestrial: Science and Alien |> >Intelligence,] (E. Regis, ed.) Cambridge University Press, 1985. This |> >is a cute theory based on some experiments with very small Turing |> >machines. It turned out that many of them performed operations that |> >could be interpreted as elementary addition -- while none of them did |> >anything that was "similar" to addition but not exactly addition! How do you define "similar" to? Do you mean any associative property with inverses and an identity? Any accumlator? It seems that there will be no woman similar to Mom because we known Mom so well ... but someone for whom she isn't the point of reference may very well see her as similar to her sister. And as addition is all that we know, and we know it so well, what precautions did you take to prevent the conclusion from following directly from your point of reference? |> >What that seems to mean is that the most elementary mathematics -- or, |> >rather, the kinds that humans have historically first imagined -- hold |> >a peculiar position among "all possible mathematical systems". In a |> >sense, they might simply be the ones that are "easiest for a machine |> >to think of". |> |> Do you mean "easiest for a human to think of"? If they're so simple, |> why has it taken us so long to develop them? It wasn't until the |> renaissance (sp?) that anyone really thought of modelling the world |> in mathematical terms alone. People must be indoctrinated for years |> before the conventions of math and physics seem 'natural'. But these thousands of years still produced the first results. It took even *longer* to develop alternatives, and will take longer still to develop alternatives that we haven't come up with yet. Simplest doesn't mean simple, and the Universe has had some time to work. |> > Why, then, might they help in making physics theories? |> >Either because the universe, too, is peculiarly simple -- whatever |> >that means -- or that the simplest theories are (at least, at first) |> >the most useful ones -- simply because they are the first ones we can |> >use to make any predictions at all... |> |> This sounds somewhat too idealized to me. The development of any kind |> of theories historically have been largely done by adapting old ideas |> and conventions, even where they are out of context (analogy). This |> implies more of a feedback mechanism of conception on conception than |> a progression of theories corrected by regular reference to "the |> universe" as you seem to be saying. The former would be much more |> haphazard and less 'logical' than than the second (and take much more |> time and arguement), which seems to agree better with the actual |> history of the sciences. :> In other words, the way we look at the |> world is perhaps more arbitrary than you think - although I cannot |> produce any evidence to support this, not having a sufficiently |> non-human perspective to refer to! But what does it take to find the original state that we use as the basis of an analogy? It is easier to understand the front wheel of a tricycle than of a car, because more of it is exposed. And perhaps mathematics is more "exposed" than the alternatives. -jJ jjewett@math.lsa.umich.edu Take only memories. Jewett@ub.cc.umich.edu Leave not even footprints.
minsky@media-lab.MEDIA.MIT.EDU (Marvin Minsky) (10/08/90)
In article <1990Oct7.192212.24550@math.lsa.umich.edu> jjewett@math.lsa.umich.edu (Jim Jewett) writes: >|> In article <3549@media-lab.MEDIA.MIT.EDU> >minsky@media-lab.media.mit.edu (Marvin Minsky) writes: >>(Minsky) ...in an essay of mine --- "Communication with Alien Intelligence," (is) a cute theory based on some experiments with very small Turing machines. It turned out that many of them performed operations that could be interpreted as elementary addition -- while none of them did anything that was "similar" to addition but not exactly addition! >How do you define "similar" to? Do you mean any associative property >with inverses and an identity? ny accumlator? It seems that there >will be no woman similar to Mom because we known Mom so well ... but >someone for whom she isn't the point of reference may very well see >her as similar to her sister. Consider reading the paper. By "similar" I meant commonsensical things -- like could there be anything like the integers except skipping the number 5? Or could there be a number system with the symmetry laws holding usually but not always. Or integers with tree signs, instead of two, etc.
cpshelley@violet.uwaterloo.ca (cameron shelley) (10/10/90)
In article <1990Oct7.192212.24550@math.lsa.umich.edu> jjewett@math.lsa.umich.edu (Jim Jewett) writes: >In article <1990Oct4.175806.7711@watdragon.waterloo.edu>, >cpshelley@violet.uwaterloo.ca (cameron shelley) writes: >|> In article <3549@media-lab.MEDIA.MIT.EDU> >minsky@media-lab.media.mit.edu (Marvin Minsky) writes: > >|> > But there is another possibility >|> >explained in an essay of mine --- "Communication with Alien >|> >Intelligence," in @i[Extraterrestrial: Science and Alien >|> >Intelligence,] (E. Regis, ed.) Cambridge University Press, 1985. This >|> >is a cute theory based on some experiments with very small Turing >|> >machines. It turned out that many of them performed operations that >|> >could be interpreted as elementary addition -- while none of them did >|> >anything that was "similar" to addition but not exactly addition! > >How do you define "similar" to? Do you mean any associative property >with inverses and an identity? Any accumlator? It seems that there >will be no woman similar to Mom because we known Mom so well ... but >someone for whom she isn't the point of reference may very well see >her as similar to her sister. > >And as addition is all that we know, and we know it so well, what precautions >did you take to prevent the conclusion from following directly from your >point of reference? > >|> >What that seems to mean is that the most elementary mathematics -- or, >|> >rather, the kinds that humans have historically first imagined -- hold >|> >a peculiar position among "all possible mathematical systems". In a >|> >sense, they might simply be the ones that are "easiest for a machine >|> >to think of". >|> >|> Do you mean "easiest for a human to think of"? If they're so simple, >|> why has it taken us so long to develop them? It wasn't until the >|> renaissance (sp?) that anyone really thought of modelling the world >|> in mathematical terms alone. People must be indoctrinated for years >|> before the conventions of math and physics seem 'natural'. > >But these thousands of years still produced the first results. It took >even *longer* to develop alternatives, and will take longer still to >develop alternatives that we haven't come up with yet. Simplest doesn't >mean simple, and the Universe has had some time to work. > Yes, we have results. But I suspect that our established "point of reference" has effectively biased our view of history in this matter. I would rather attribute the success of a particular alternative to a more random process of precedence than some ill-defined (or worse circularly defined) 'innateness'. The success of one view, once widely accepted has traditionally retarded other views from being seriously examined, despite simlicity, expressiveness, and indeed priority. Our knowledge of what was actually going on in the Ancient mathematical world has been written by the winners of the competition, the views of the losers suppressed or just plain lost. In other words, there is more human psychology at work here than mere perception of simplicity. >|> > Why, then, might they help in making physics theories? >|> >Either because the universe, too, is peculiarly simple -- whatever >|> >that means -- or that the simplest theories are (at least, at first) >|> >the most useful ones -- simply because they are the first ones we can >|> >use to make any predictions at all... >|> >|> This sounds somewhat too idealized to me. The development of any kind >|> of theories historically have been largely done by adapting old ideas >|> and conventions, even where they are out of context (analogy). This >|> implies more of a feedback mechanism of conception on conception than >|> a progression of theories corrected by regular reference to "the >|> universe" as you seem to be saying. The former would be much more >|> haphazard and less 'logical' than than the second (and take much more >|> time and arguement), which seems to agree better with the actual >|> history of the sciences. :> In other words, the way we look at the >|> world is perhaps more arbitrary than you think - although I cannot >|> produce any evidence to support this, not having a sufficiently >|> non-human perspective to refer to! > >But what does it take to find the original state that we use as the basis >of an analogy? It is easier to understand the front wheel of a tricycle >than of a car, because more of it is exposed. And perhaps mathematics >is more "exposed" than the alternatives. > Perhaps, but to make that a definite assertion is begging the question. Whatever really went on in history and prehistory is very obscure, and that is not liable to change much. -- Cameron Shelley | "Saw, n. A trite popular saying, or proverb. cpshelley@violet.waterloo.edu| So called because it makes its way into a Davis Centre Rm 2136 | wooden head." Phone (519) 885-1211 x3390 | Ambrose Bierce