[math-fun] Definition of mathematics
I still like mine better than everybody else's: it is the study of thinking & how to be a better thinker. Is it the study of structure, patterns & form? No, because it also encompasses the study of phenomena that seem structureless and formless. How about chaos, fluid turbulence, and random number generators? Are they to be excluded from mathematics? Is it algebraic & symbolic manipulations? No, that would exclude much ancient greek geometry-based mathematics. Is it the study of "numbers" and their descendants? How about set theory, topology, and logic as counterexamples? Is it physics? Well, plenty of maths has nothing to do with physics, in fact sometimes intentionally so, such as the study of laws intentionally designed to be wrong-physics. Is it the study of tautologies and obviously-true statements? Dubious, counterexamples have already been posted. Is it the study of rule-sets we make up (Vi Hart)? Not a bad try, that one. I think all those definitions were trying to be my definition but falling a little short, except for maybe Vi Hart's definition which arguably is equivalent to mine. So there :) On 7/6/15, math-fun-request@mailman.xmission.com <math-fun-request@mailman.xmission.com> wrote:
Send math-fun mailing list submissions to math-fun@mailman.xmission.com
To subscribe or unsubscribe via the World Wide Web, visit https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun or, via email, send a message with subject or body 'help' to math-fun-request@mailman.xmission.com
You can reach the person managing the list at math-fun-owner@mailman.xmission.com
When replying, please edit your Subject line so it is more specific than "Re: Contents of math-fun digest..."
Today's Topics:
1. Re: Definition of mathematics (meekerdb) 2. Re: Definition of mathematics (James Propp) 3. Re: Definition of mathematics (James Propp) 4. Re: Definition of mathematics (meekerdb) 5. Re: New(?) way to tabulate pFq identities (Bill Gosper)
----------------------------------------------------------------------
Message: 1 Date: Sun, 05 Jul 2015 18:28:11 -0700 From: meekerdb <meekerdb@verizon.net> To: math-fun <math-fun@mailman.xmission.com> Subject: Re: [math-fun] Definition of mathematics Message-ID: <5599D9AB.9080209@verizon.net> Content-Type: text/plain; charset=utf-8; format=flowed
I can add a few:
"The duty of abstract mathematics, as I see it, is precisely to expand our capacity for hypothesizing possible ontologies." --- Norm Levitt
A physicist goes off to a conference. After a week his suit?s gotten soiled and crumpled, so he goes out to look for a dry cleaner. Walking down the main street of town, he comes upon a store with a lot of signs out front. One of them says ?Dry Cleaning.? So he goes in with his dirty suit and asks when he can come back to pick it up. The mathematician who owns the shop replies, ?I?m terribly sorry, but we don?t do dry cleaning.? ?What?? exclaims the puzzled physicist. ?The sign outside says ?Dry Cleaning?!? ?We do not do anything here,? replies the mathematician. ?We only sell signs!? --- Alain Connes, in Changeux
"A mathematician is like a mad tailor: he is making "all possible clothes" and hopes to make also something suitable for dressing" --- Stanislaw Lem, Summa Techologiae
"Mathematics is part of physics. Physics is an experimental science, a part of natural science. Mathematics is the part of physics where experiments are cheap." --- Vladimir Arnold.
"Math is a cybervirus that lives in human minds, evolves therein and reproduces itself via language." --- Stephen Paul King
Brent Meeker
On 7/5/2015 2:29 PM, Thane Plambeck wrote:
i just noticed there are several proposed definitions (some of them familiar to me, others not), on the wikipedia page for bertrand russell, and dropped in below
Bertrand Russell wrote this famous tongue-in-cheek definition, describing the way all terms in mathematics are ultimately defined by reference to undefined terms:
The subject in which we never know what we are talking about, nor whether what we are saying is true.[12] <https://en.m.wikipedia.org/wiki/Definitions_of_mathematics#cite_note-12> Bertrand Russell <https://en.m.wikipedia.org/wiki/Bertrand_Russell> 1901
Many other attempts to characterize mathematics have led to humor or poetic prose:
"Mathematics is about making up rules and seeing what happens."[13] <https://en.m.wikipedia.org/wiki/Definitions_of_mathematics#cite_note-13> Vi Hart <https://en.m.wikipedia.org/wiki/Vi_Hart>
A mathematician is a blind man in a dark room looking for a black cat which isn't there.[14] <https://en.m.wikipedia.org/wiki/Definitions_of_mathematics#cite_note-14> Charles Darwin <https://en.m.wikipedia.org/wiki/Charles_Darwin>
A mathematician, like a painter or poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas. G. H. Hardy <https://en.m.wikipedia.org/wiki/G._H._Hardy>, 1940
Mathematics is the art of giving the same name to different things.[8] <https://en.m.wikipedia.org/wiki/Definitions_of_mathematics#cite_note-eves-8> Henri Poincar? <https://en.m.wikipedia.org/wiki/Henri_Poincar%C3%A9>
Mathematics is the science of skilful operations with concepts and rules invented just for this purpose. [this purpose being the skilful operation ....][15] <https://en.m.wikipedia.org/wiki/Definitions_of_mathematics#cite_note-15> Eugene Wigner <https://en.m.wikipedia.org/wiki/Eugene_Wigner>
Mathematics is not a book confined within a cover and bound between brazen clasps, whose contents it needs only patience to ransack; it is not a mine, whose treasures may take long to reduce into possession, but which fill only a limited number of veins and lodes; it is not a soil, whose fertility can be exhausted by the yield of successive harvests; it is not a continent or an ocean, whose area can be mapped out and its contour defined: it is limitless as that space which it finds too narrow for its aspirations; its possibilities are as infinite as the worlds which are forever crowding in and multiplying upon the astronomer's gaze; it is as incapable of being restricted within assigned boundaries or being reduced to definitions of permanent validity, as the consciousness of life, which seems to slumber in each monad, in every atom of matter, in each leaf and bud cell, and is forever ready to burst forth into new forms of vegetable and animal existence.[16] <https://en.m.wikipedia.org/wiki/Definitions_of_mathematics#cite_note-StewartFrom-16> James Joseph Sylvester <https://en.m.wikipedia.org/wiki/James_Joseph_Sylvester>
What is mathematics? What is it for? What are mathematicians doing nowadays? Wasn't it all finished long ago? How many new numbers can you invent anyway? Is today's mathematics just a matter of huge calculations, with the mathematician as a kind of zookeeper, making sure the precious computers are fed and watered? If it's not, what is it other than the incomprehensible outpourings of superpowered brainboxes with their heads in the clouds and their feet dangling from the lofty balconies of their ivory towers? Mathematics is all of these, and none. Mostly, it's just different. It's not what you expect it to be, you turn your back for a moment and it's changed. It's certainly not just a fixed body of knowledge, its growth is not confined to inventing new numbers, and its hidden tendrils pervade every aspect of modern life.[16] <https://en.m.wikipedia.org/wiki/Definitions_of_mathematics#cite_note-StewartFrom-16> Ian Stewart <https://en.m.wikipedia.org/wiki/Ian_Stewart_(mathematician)>
On Sunday, July 5, 2015, Andy Latto <andy.latto@pobox.com> wrote:
While you and I may both agree that the set of things that we couldn't imagine were otherwise roughly corresponds to mathematics, there are a lot of theological arguments that porport to prove that there is a God, and that he has certain properties, and that this couldn't possibly be otherwise. So I don't see a good way to patch this definition to exclude theology.
Andy
On Sat, Jul 4, 2015 at 3:30 PM, James Propp <jamespropp@gmail.com <javascript:;>> wrote:
Here's a start of a different sort of approach to the question of what math is, taking off from a remark in Jordan Ellenberg's book "How Not To Be Wrong" about how a times b equals b times a, for all a and b, "because it couldn't be otherwise".
Let's make this more modest and say that we can't IMAGINE how it could be otherwise.
That is: I can just barely imagine (with a mental squint, and with an inner acknowledgments of my limitations as a reasoner) *that* ordinary multiplication of ordinary natural numbers might not be commutative. But I cannot imagine in any kind of detail *how* it might fail to be commutative. There are probably lots of things that humans aren't able to doubt (such as "I exist") that don't count as mathematics, so this definition will need to be modified before it comes close to drawing the line between math and non-math in approximately the right place.
A variant of this approach would be to define pure mathematics as the study of fantasies that possess a certain kind of coherence.
Jim Propp _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com <javascript:;> https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
-- Andy.Latto@pobox.com <javascript:;>
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com <javascript:;> https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
------------------------------
Message: 2 Date: Sun, 5 Jul 2015 22:02:13 -0400 From: James Propp <jamespropp@gmail.com> To: math-fun <math-fun@mailman.xmission.com> Subject: Re: [math-fun] Definition of mathematics Message-ID: <CA+G9J-cAFDyFBFzVBxMH0r=Ao_hNtF_08knNd88evmPAjWmoFA@mail.gmail.com> Content-Type: text/plain; charset=UTF-8
Thanks, Brent.
I'll comment on a couple of these:
"The duty of abstract mathematics, as I see it, is precisely to
expand our capacity for hypothesizing possible ontologies."
I'll combine this with quotes from J. B. S. Haldane ("*Now my own suspicion is that the Universe is not only queerer than we suppose, but queerer than we can suppose*") and Francis Bacon (the end goal of science is "the effecting of all things possible") and assert that the job of the pure mathematician is the imagining of all things possible, however queer.
"Math is a cybervirus that lives in human minds, evolves therein and
reproduces itself via language."
Here's my more specific analysis of the memetic ecology of math (this time based on a quote that is often credited to Samuel Butler): A theorem is only a question's way of making more questions.
Jim Propp
------------------------------
Message: 3 Date: Sun, 5 Jul 2015 22:26:25 -0400 From: James Propp <jamespropp@gmail.com> To: math-fun <math-fun@mailman.xmission.com> Subject: Re: [math-fun] Definition of mathematics Message-ID: <CA+G9J-fqhhuOEzohQFzw3Jds5ssyTEv_=DR=Kfb2AqtANZho6A@mail.gmail.com> Content-Type: text/plain; charset=UTF-8
Is the quote from Levitt part of an essay that elaborates on this point, and if so, where can I find it?
"The duty of abstract mathematics, as I see it, is precisely to
expand our capacity for hypothesizing possible ontologies." --- Norm Levitt
Jim Propp
------------------------------
Message: 4 Date: Sun, 05 Jul 2015 22:11:00 -0700 From: meekerdb <meekerdb@verizon.net> To: math-fun <math-fun@mailman.xmission.com> Subject: Re: [math-fun] Definition of mathematics Message-ID: <559A0DE4.4070201@verizon.net> Content-Type: text/plain; charset=windows-1252; format=flowed
No, it's "private communication" as he and I both posted on Vic Stenger's email list. Sadly both Vic and Norm have died.
Brent Meeker
On 7/5/2015 7:26 PM, James Propp wrote:
Is the quote from Levitt part of an essay that elaborates on this point, and if so, where can I find it?
"The duty of abstract mathematics, as I see it, is precisely to
expand our capacity for hypothesizing possible ontologies." --- Norm Levitt
Jim Propp _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
------------------------------
Message: 5 Date: Sun, 5 Jul 2015 22:27:04 -0700 From: Bill Gosper <billgosper@gmail.com> To: math-fun@mailman.xmission.com Subject: Re: [math-fun] New(?) way to tabulate pFq identities Message-ID: <CAA-4O0H6Uj=yzcWJ+hZc8ZGNuCsuLmP1Xp6QCZe5xnAg1BRSmg@mail.gmail.com> Content-Type: text/plain; charset=UTF-8
On Tue, Jun 30, 2015 at 12:25 PM, Bill Gosper <billgosper@gmail.com> wrote:
By a process not obvious how to automate,
In very rough and somewhat gory form: gosper.org/hyper3by3.pdf (Pleasant introduction: gosper.org/pathi.pdf). Included: a triple 4F3[+1/4] = double 5F4 = single 6f5 (with conjugate quadratic surd parameters) all with three freedoms, as a subproblem. Probably nobody wants quadratic surd parameters, but there are numerous ways of rationalizing them by surrendering one freedom. So this is a compact way of transmitting them. A CAS could either exhaustively tabulate all the rational specializations, or just keep the general form, and check for a matching query by solving algebraic equations on the fly. This might be pushing the envelope when we get to cubics and beyond. --rwg
I extracted
HypergeometricPFQ[{a, b, (2 a)/5 + (6 b)/5 + 1, 2 a + 2 b, a + 3 b}, {(2 a)/5 + (6 b)/5, a + (3 b)/2 + 1/2, a + (3 b)/2 + 1, 2 b + 1}, -(1/4)] == (4^-a Gamma[b + 1/2] Gamma[2 a + 3 b + 1])/( Gamma[a + b + 1/2] Gamma[a + 3 b + 1])
from a summation containing a messy, irreducible cubic factor, analogous to the sort used to discover recurrence relations via "creative telescopy". Punting the cubic cost two degrees of freedom, but there were hundreds of variants. The unpleasant choice seems to be between a useless messy cubic, and a nearly useless identity with only two degrees of freedom. Probably the identity you need was among the hundreds I threw away.
Here's an interesting remedy. What you actually tabulate is, in this case, the fully general result in a somewhat intimidating
nauseating
form:
((-b - d) ((1/(-1 + b))(-1 + b - d) (-1 + a + b - d) (-1 + b + c - d) HypergeometricPFQ[{a, -1 + b, c, d, -1 - a + b + d, -1 + b - c + d}, {b/2 + d/2,
[This was so miscanonicalized that I'm vandalizing it here to make sure nobody wastes time trying to use it.]
It's then fairly routine to determine whether your pFq[-1/4] is a special
case.
I.e., you check your z argument = -1/4 and your parameters are congruent mod1.
It's not pencil and paper, but many people now have access to computers. --rwg
OK, here it is cleaned up.
-(b - 2 d) (a + b - 2 d) (b + c - 2 d) HypergeometricPFQ[{a, -a + b, b - c, c, b - d, d}, {1/2 + b/2, 1 + b/2, 1 - a + b - c, 1 - a + d, 1 - c + d}, -(1/4)] + (b - d) (5 + 3 c + b (9 + 5 b + 4 c) + a (3 + 4 b + c - 7 d) - 18 d - 7 (3 b + c) d + 21 d^2) HypergeometricPFQ[{a, -a + b, b - c, c, 1 + b - d, d}, {1/2 + b/2, 1 + b/2, 1 - a + b - c, 1 - a + d, 1 - c + d}, -(1/4)] - 3 (5 + a + 3 b + c - 6 d) (b - d) (1 + b - d) HypergeometricPFQ[{a, -a + b, b - c, c, 2 + b - d, d}, {1/2 + b/2, 1 + b/2, 1 - a + b - c, 1 - a + d, 1 - c + d}, -( 1/4)] + 5 (b - d) (1 + b - d) (2 + b - d) HypergeometricPFQ[{a, -a + b, b - c, c, 3 + b - d, d}, {1/2 + b/2, 1 + b/2, 1 - a + b - c, 1 - a + d, 1 - c + d}, -(1/4)] == ( Gamma[1 + b] Gamma[1 - a + b - c] Gamma[1 - a + d] Gamma[1 - c + d])/ (Gamma[-a + b] Gamma[b - c] Gamma[d] Gamma[1 - a - c + d])
But maybe instead of a special case, you have a limiting case, say a -> oo :
(b - 2 d) (b + c - 2 d) HypergeometricPFQ[{b - c, c, b - d, d}, {1/2 + b/2, 1 + b/2, 1 - c + d}, 1/ 4] - (3 + 4 b + c - 7 d) (b - d) HypergeometricPFQ[{b - c, c, 1 + b - d, d}, {1/2 + b/2, 1 + b/2, 1 - c + d}, 1/4] + 3 (b - d) (1 + b - d) HypergeometricPFQ[{b - c, c, 2 + b - d, d}, {1/2 + b/2, 1 + b/2, 1 - c + d}, 1/4] == ( Gamma[1 + b] Gamma[1 - c + d])/(Gamma[b - c] Gamma[d])
These are now 4F3[+1/4] instead of 6F5[-1/4] ! Also note the rhs of the a->oo limit is not the a->oo limit of the 6F5 rhs, but rather that limit transformed by Gamma[z]->1/Gamma[1-z] . I have only the vaguest notion why. Maybe there were illegal limit interchanges when I blew up my contour rectangles.
I'm a big fan of handling all the contiguous cases, although it seems ambitious. I think for something like this you'd need a thirteen(?) dimensional system of 7x7 matrices, and seven-term recurrences. The chance that your 6F5[-1/4] will come out in Gammas seems mighty slim. Slightly better for certain 7F6[a+1,b,...g; a,h,...,i,j,k,L|-1/4] --rwg (Some of you may wonder about Mathematica's line-breaking. It's deliberately ghastly, to preclude premature end-of-expression on readback.)
------------------------------
Subject: Digest Footer
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
------------------------------
End of math-fun Digest, Vol 149, Issue 17 *****************************************
-- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)
In my view, mathematics is definitely *not* the study of thinking. There are huge areas in cognitive psychology for that and mathematics exists irrespective of one's level of thinking (yes, my philosophy is showing). I lean toward the view that math is the study of structure, patterns, and form that arise as a consequence of rule sets. The things that Warren listed that *seem* formless are not actually. Chaos has form; fluid turbulence has structure. Even truly random events & sets have structure that can be studied mathematically. Kerry On Mon, Jul 6, 2015 at 9:30 AM, Warren D Smith <warren.wds@gmail.com> wrote:
I still like mine better than everybody else's: it is the study of thinking & how to be a better thinker.
Is it the study of structure, patterns & form? No, because it also encompasses the study of phenomena that seem structureless and formless. How about chaos, fluid turbulence, and random number generators? Are they to be excluded from mathematics? Is it algebraic & symbolic manipulations? No, that would exclude much ancient greek geometry-based mathematics. Is it the study of "numbers" and their descendants? How about set theory, topology, and logic as counterexamples? Is it physics? Well, plenty of maths has nothing to do with physics, in fact sometimes intentionally so, such as the study of laws intentionally designed to be wrong-physics. Is it the study of tautologies and obviously-true statements? Dubious, counterexamples have already been posted. Is it the study of rule-sets we make up (Vi Hart)? Not a bad try, that one. I think all those definitions were trying to be my definition but falling a little short, except for maybe Vi Hart's definition which arguably is equivalent to mine.
So there :)
On Mon, Jul 6, 2015 at 9:30 AM, Warren D Smith <warren.wds@gmail.com> wrote:
Is it the study of "numbers" and their descendants? How about set theory, topology, and logic as counterexamples?
I wouldn't consider those counterexamples. Set theory started with finite sets; Aristotle argued that natural numbers were only "potentially" infinite, not "actually" infinite. Topology started with the bridges of Konigsberg, Euler's formula V-E+F=2, and Lhuilier's generalization to polyhedra with higher genus V-E+F=2-2g. I'd say that logic started with bookkeeping (a lovely word with three consecutive double letters) and the rules of deduction in arithmetic. -- Mike Stay - metaweta@gmail.com http://www.cs.auckland.ac.nz/~mike http://reperiendi.wordpress.com
On 7/6/2015 9:30 AM, Warren D Smith wrote:
I still like mine better than everybody else's: it is the study of thinking & how to be a better thinker.
Depends on what you mean by better. In many circumstances it is more important to be quick than to prove inferences. That's why evolution didn't produced a lot of logicians. Brent Meeker
There are several different mathematicses that would each have its own definition, including at least the following: 1) The human culture of any field that uses mathematics to study something 2) The human culture of mathematicians studying pure math (the study of deriving rigorous results from rigorous rules*) 3) The study of deriving rigorous results from rigorous rules per se 4) The body of past, present, future rigorous results from rigorous rules 5) The body of past, present, future rigorous results from axiom schema of any cardinality This last one besides being interesting in its own right includes, e.g., all exactly definition of the integers, satisfying Platonists of all stripes —Dan ____________________________________________________________________ * "Here "rigorous rules" meaning a finite collection of axiom schema
If you make inferences that are truth-preserving, precise, and consistent you're doing mathematics. Brent Meeker On 7/6/2015 11:09 AM, Dan Asimov wrote:
There are several different mathematicses that would each have its own definition, including at least the following:
1) The human culture of any field that uses mathematics to study something
2) The human culture of mathematicians studying pure math (the study of deriving rigorous results from rigorous rules*)
3) The study of deriving rigorous results from rigorous rules per se
4) The body of past, present, future rigorous results from rigorous rules
5) The body of past, present, future rigorous results from axiom schema of any cardinality
This last one besides being interesting in its own right includes, e.g., all exactly definition of the integers, satisfying Platonists of all stripes
—Dan
____________________________________________________________________ * "Here "rigorous rules" meaning a finite collection of axiom schema _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
On Mon, Jul 6, 2015 at 2:09 PM, Dan Asimov <asimov@msri.org> wrote:
There are several different mathematicses that would each have its own definition, including at least the following:
1) The human culture of any field that uses mathematics to study something
2) The human culture of mathematicians studying pure math (the study of deriving rigorous results from rigorous rules*)
Unless you want to exclude Gauss, Euler, and Euclid from this culture, I think you need a different definition (or a slightly different mathematics 2A, that also requires its own definition) that includes the study of the same subject in ways that do not meet modern standards of rigor. Andy
3) The study of deriving rigorous results from rigorous rules per se
4) The body of past, present, future rigorous results from rigorous rules
5) The body of past, present, future rigorous results from axiom schema of any cardinality
This last one besides being interesting in its own right includes, e.g., all exactly definition of the integers, satisfying Platonists of all stripes
—Dan
____________________________________________________________________ * "Here "rigorous rules" meaning a finite collection of axiom schema _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
-- Andy.Latto@pobox.com
We're pretty far from perfect standards of rigor even now, with set theory on shaky foundations, and lots of topological arguments relying on pictures rather than exact reasoned arguments. Still, it is all the human study of deriving rigorous results — even if we still don't know exactly how to do this. —Dan
On Jul 6, 2015, at 1:08 PM, Andy Latto <andy.latto@pobox.com> wrote:
2) The human culture of mathematicians studying pure math (the study of deriving rigorous results from rigorous rules*)
Unless you want to exclude Gauss, Euler, and Euclid from this culture, I think you need a different definition (or a slightly different mathematics 2A, that also requires its own definition) that includes the study of the same subject in ways that do not meet modern standards of rigor.
On Monday, July 6, 2015, Warren D Smith <warren.wds@gmail.com> wrote: I still like mine better than everybody else's: it is the study of
thinking & how to be a better thinker.
Graduate students in the humanities are learning how to think more intelligently in their chosen specialties. The same goes for lawyers. But they're not doing mathematics. (I omit medicine and engineering and the sciences, since they have significant overlap with mathematics: some of the thinking skills one acquires in these fields are mathematical, and others aren't.) Jim Propp
participants (7)
-
Andy Latto -
Dan Asimov -
James Propp -
Kerry Mitchell -
meekerdb -
Mike Stay -
Warren D Smith