The Utility of Mathematics(catb.org)
catb.org
The Utility of Mathematics
http://www.catb.org/~esr/writings/utility-of-math/
4 comments
And Richard Hamming's follow up in 1980:
http://www-lmmb.ncifcrf.gov/~toms/Hamming.unreasonable.html
I didn't know about this, thanks!
While we are at it, there is also the 2009 paper: The Unreasonable Effectiveness of Data by Peter Norvig.
While we are at it, there is also the 2009 paper: The Unreasonable Effectiveness of Data by Peter Norvig.
ESR's insight is almost trivial on the surface, but reveals a deep understanding in the same sort of way Eliezer Yudkowsky demystifies most nonsense philosophical questions.
http://lesswrong.com/lw/no/how_an_algorithm_feels_from_insid... http://lesswrong.com/lw/of/dissolving_the_question/
The real problem to address is not finding the answer to "mysterious" questions, but understanding the algorithm that makes our brain categorize those questions as such.
I have had somewhat the same realization ESR gives here, but never so transparently elucidated. With no irony, I translate the statement he addresses into a mathematical like nature: "Out of formal models capable of giving predictive powers to informal models, there exists a (unique?) maximally such predictive model."
Proving the maximality is then easy if we simply define "maximally predictive" to be "best we've done so far!" (I do not think it is an interesting question to ask whether there is some sort of fuzzy idea of partial order on predictive power, or if this partial order has multiple maximal elements.)
The existence statement is the real beautiful reduction, to me, showing the absurdity of the original question. What ESR describes to me is precisely a kind of selection bias; we ignore talking about "predictive formal models" for all the situations mathematics has not come galloping in as saviour!
This latter argument is eerily similar to one of the arguments given forth by Dawkins when I read his The God Delusion as a child and turned atheist. Namely, the idea that I was already an atheist to every other religion out there, so why not go one further? The power of such arguments to actually sway and rethink my opinion as an advertised rationalist makes me tremble in their powers, curious whether this effectiveness is deserved, or somehow a way to hack a rationalist's brain.
I feel and believe ESR answered reasonably well the question he also posed fairly well, but my mind can't help screaming "Is an almost trivial reformulation really all there is to it?!" This is not so much my expression of doubt as of awe.
http://lesswrong.com/lw/no/how_an_algorithm_feels_from_insid... http://lesswrong.com/lw/of/dissolving_the_question/
The real problem to address is not finding the answer to "mysterious" questions, but understanding the algorithm that makes our brain categorize those questions as such.
I have had somewhat the same realization ESR gives here, but never so transparently elucidated. With no irony, I translate the statement he addresses into a mathematical like nature: "Out of formal models capable of giving predictive powers to informal models, there exists a (unique?) maximally such predictive model."
Proving the maximality is then easy if we simply define "maximally predictive" to be "best we've done so far!" (I do not think it is an interesting question to ask whether there is some sort of fuzzy idea of partial order on predictive power, or if this partial order has multiple maximal elements.)
The existence statement is the real beautiful reduction, to me, showing the absurdity of the original question. What ESR describes to me is precisely a kind of selection bias; we ignore talking about "predictive formal models" for all the situations mathematics has not come galloping in as saviour!
This latter argument is eerily similar to one of the arguments given forth by Dawkins when I read his The God Delusion as a child and turned atheist. Namely, the idea that I was already an atheist to every other religion out there, so why not go one further? The power of such arguments to actually sway and rethink my opinion as an advertised rationalist makes me tremble in their powers, curious whether this effectiveness is deserved, or somehow a way to hack a rationalist's brain.
I feel and believe ESR answered reasonably well the question he also posed fairly well, but my mind can't help screaming "Is an almost trivial reformulation really all there is to it?!" This is not so much my expression of doubt as of awe.
This is the best essay on this subject I've ever read.
I have had some thoughts on why mathematics is so fundamental to nature, but am not sure if these make any sense, or if they are ultimately cyclic.
I use words "stuff" and "things" to refer to not specifically matter but just any concept that may exist in the universe, maybe even just space or time.
1. The universe may comprise stuff that does not interact with other stuff in any meaningful way. (One issue here is that by merely talking about stuff, and more stuff, at least some aspects of set theory have already been assumed. More on this below.) (Such stuff would be unobservable to lifeforms within the universe so thinking about such stuff is meaningless for us.)
2. I then invoke something like Plenitude principle to conclude that there are things in the universe that interact with each other.
3. Interaction between any two things in the universe, is fundamentally same as transfer of some information between the two. In other words, these two things appear different to us only because our language has two different ways of referring to this.
4. Information is meaningless if it is completely random. So there must be some pattern to it, or else #3 above is meaningless. (In other words, it must be possible to describe the information transferred with less "bits" than the amount of "bits" transferred. Or the Kolmogorov complexity must be less than the amount of information transferred.) (While I use mathematical terms to convey the message, it is just because I need a language to say this.)
5. Existence of any pattern whatsoever in information transferred leads to mathematics that we then define to describe the pattern.
Note that this model readily allows for:
A. Laws of physics that are probabilistic in nature. The information transfer must not be completely random, but this of course does not require it to be fully deterministic.
B. Stuff that is unobservable or very weakly observable. Dark matter, neutrinos, etc. would be examples of such stuff.
The biggest issue I know of that remains is assumption of at least some axioms of set theory, which also is a foundation of mathematics.
There could be universe(s) where there is either none or at most one of anything/everything. Such a universe again would not have anything observable and is thus meaningless to us. Again invoking Plenitude principle, I assume existence of a universe where there are more than one of something/anything. But the moment one thinks of two things (say two particles, or even two values of a scalar variable like time or position), aspects of set theory or number theory have already come in.
On the other hand, and I believe for the same reason, mathematicians are unable to define what is meant by a set or a membership to a set. (Everything I have read on set theory starts by assuming some meaning for these two terms and then follows with the rest.)
Would love to get feedback from others on if this makes sense, or if this is fundamentally wrong or cyclic.
Reason for edits: Fixed some typos.
I use words "stuff" and "things" to refer to not specifically matter but just any concept that may exist in the universe, maybe even just space or time.
1. The universe may comprise stuff that does not interact with other stuff in any meaningful way. (One issue here is that by merely talking about stuff, and more stuff, at least some aspects of set theory have already been assumed. More on this below.) (Such stuff would be unobservable to lifeforms within the universe so thinking about such stuff is meaningless for us.)
2. I then invoke something like Plenitude principle to conclude that there are things in the universe that interact with each other.
3. Interaction between any two things in the universe, is fundamentally same as transfer of some information between the two. In other words, these two things appear different to us only because our language has two different ways of referring to this.
4. Information is meaningless if it is completely random. So there must be some pattern to it, or else #3 above is meaningless. (In other words, it must be possible to describe the information transferred with less "bits" than the amount of "bits" transferred. Or the Kolmogorov complexity must be less than the amount of information transferred.) (While I use mathematical terms to convey the message, it is just because I need a language to say this.)
5. Existence of any pattern whatsoever in information transferred leads to mathematics that we then define to describe the pattern.
Note that this model readily allows for:
A. Laws of physics that are probabilistic in nature. The information transfer must not be completely random, but this of course does not require it to be fully deterministic.
B. Stuff that is unobservable or very weakly observable. Dark matter, neutrinos, etc. would be examples of such stuff.
The biggest issue I know of that remains is assumption of at least some axioms of set theory, which also is a foundation of mathematics.
There could be universe(s) where there is either none or at most one of anything/everything. Such a universe again would not have anything observable and is thus meaningless to us. Again invoking Plenitude principle, I assume existence of a universe where there are more than one of something/anything. But the moment one thinks of two things (say two particles, or even two values of a scalar variable like time or position), aspects of set theory or number theory have already come in.
On the other hand, and I believe for the same reason, mathematicians are unable to define what is meant by a set or a membership to a set. (Everything I have read on set theory starts by assuming some meaning for these two terms and then follows with the rest.)
Would love to get feedback from others on if this makes sense, or if this is fundamentally wrong or cyclic.
Reason for edits: Fixed some typos.
http://en.wikipedia.org/wiki/The_Unreasonable_Effectiveness_...