[math-fun] Definition of mathematics
A couple of weeks ago, in a gathering of mathematicians throwing out candidate definitions of mathematics, I half-seriously ventured the opinion that mathematics is the subset of philosophy consisting of those philosophical questions that actually have answers, along with the answers to those questions. But I don't think this is original. Whom am I quoting (or paraphrasing) here? Jim Propp
Mathematics is the study of the formal consequences of formal rules. Even more succinctly, mathematics is the study of form. On Thu, Jun 11, 2015 at 6:23 PM, James Propp <jamespropp@gmail.com> wrote:
A couple of weeks ago, in a gathering of mathematicians throwing out candidate definitions of mathematics, I half-seriously ventured the opinion that mathematics is the subset of philosophy consisting of those philosophical questions that actually have answers, along with the answers to those questions.
But I don't think this is original. Whom am I quoting (or paraphrasing) here?
Jim Propp _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
As a grad student, ca. 1971, I came up with "Mathematics is the science of patterns." —Dan
On Jun 11, 2015, at 5:13 PM, Allan Wechsler <acwacw@gmail.com> wrote:
Mathematics is the study of the formal consequences of formal rules. Even more succinctly, mathematics is the study of form.
On Thu, Jun 11, 2015 at 6:23 PM, James Propp <jamespropp@gmail.com> wrote:
A couple of weeks ago, in a gathering of mathematicians throwing out candidate definitions of mathematics, I half-seriously ventured the opinion that mathematics is the subset of philosophy consisting of those philosophical questions that actually have answers, along with the answers to those questions.
But I don't think this is original. Whom am I quoting (or paraphrasing) here?
Jim Propp _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
Gauss once said: "Mathematics is the Queen of the Sciences" Assuming we've defined `queen' and `science', and established uniqueness, we have a concrete definition of mathematics. Sincerely, Adam P. Goucher
Sent: Friday, June 12, 2015 at 2:34 AM From: "Dan Asimov" <asimov@msri.org> To: math-fun <math-fun@mailman.xmission.com> Subject: Re: [math-fun] Definition of mathematics
As a grad student, ca. 1971, I came up with
"Mathematics is the science of patterns."
—Dan
On Jun 11, 2015, at 5:13 PM, Allan Wechsler <acwacw@gmail.com> wrote:
Mathematics is the study of the formal consequences of formal rules. Even more succinctly, mathematics is the study of form.
On Thu, Jun 11, 2015 at 6:23 PM, James Propp <jamespropp@gmail.com> wrote:
A couple of weeks ago, in a gathering of mathematicians throwing out candidate definitions of mathematics, I half-seriously ventured the opinion that mathematics is the subset of philosophy consisting of those philosophical questions that actually have answers, along with the answers to those questions.
But I don't think this is original. Whom am I quoting (or paraphrasing) here?
Jim Propp _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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He went on to say that "Arithmetic is the Queen of Mathematics". What of the Kings, though? If there is one, it is Physics, but according to one historian, "Two hundred years ago, physics and we know it did not exist". So in Gauss' mind, we may presume that Mathematics was the sole reigning monarch of the sciences, each science having its own sub-queens. Perhaps he was thinking of muses, really. The idea of all the sciences being beset with lines of nobility and the resulting warring factions is disturbing. Had that happened, we'd call it the NSF. Hilarie
From: "Adam P. Goucher" <apgoucher@gmx.com> Date: Fri, 12 Jun 2015 04:37:16 +0200 Subject: Re: [math-fun] Definition of mathematics
Gauss once said:
"Mathematics is the Queen of the Sciences"
Assuming we've defined `queen' and `science', and established uniqueness, we have a concrete definition of mathematics.
Sincerely,
Adam P. Goucher
Sent: Friday, June 12, 2015 at 2:34 AM From: "Dan Asimov" <asimov@msri.org> To: math-fun <math-fun@mailman.xmission.com> Subject: Re: [math-fun] Definition of mathematics
As a grad student, ca. 1971, I came up with
"Mathematics is the science of patterns."
—Dan
On Jun 11, 2015, at 5:13 PM, Allan Wechsler <acwacw@gmail.com> wrote:
Mathematics is the study of the formal consequences of formal rules. Even more succinctly, mathematics is the study of form.
On Thu, Jun 11, 2015 at 6:23 PM, James Propp <jamespropp@gmail.com> wrote:
A couple of weeks ago, in a gathering of mathematicians throwing out candidate definitions of mathematics, I half-seriously ventured the opinion that mathematics is the subset of philosophy consisting of those philosophical questions that actually have answers, along with the answers to those questions.
But I don't think this is original. Whom am I quoting (or paraphrasing) here?
Jim Propp
He went on to say that "Arithmetic is the Queen of Mathematics". What of the Kings, though? If there is one, it is Physics, but according to one historian, "Two hundred years ago, physics and we know it did not exist". So in Gauss' mind, we may presume that Mathematics was the sole reigning monarch of the sciences, each science having its own sub-queens. Perhaps he was thinking of muses, really. The idea of all the sciences being beset with lines of nobility and the resulting warring factions is disturbing. Had that happened, we'd call it the NSF. Hilarie
From: "Adam P. Goucher" <apgoucher@gmx.com> Date: Fri, 12 Jun 2015 04:37:16 +0200 Subject: Re: [math-fun] Definition of mathematics
Gauss once said:
"Mathematics is the Queen of the Sciences"
Assuming we've defined `queen' and `science', and established uniqueness, we have a concrete definition of mathematics.
Sincerely,
Adam P. Goucher
Sent: Friday, June 12, 2015 at 2:34 AM From: "Dan Asimov" <asimov@msri.org> To: math-fun <math-fun@mailman.xmission.com> Subject: Re: [math-fun] Definition of mathematics
As a grad student, ca. 1971, I came up with
"Mathematics is the science of patterns."
Dan
On Jun 11, 2015, at 5:13 PM, Allan Wechsler <acwacw@gmail.com> wrote:
Mathematics is the study of the formal consequences of formal rules. Even more succinctly, mathematics is the study of form.
On Thu, Jun 11, 2015 at 6:23 PM, James Propp <jamespropp@gmail.com> wrote:
A couple of weeks ago, in a gathering of mathematicians throwing out candidate definitions of mathematics, I half-seriously ventured the opinion that mathematics is the subset of philosophy consisting of those philosophical questions that actually have answers, along with the answers to those questions.
But I don't think this is original. Whom am I quoting (or paraphrasing) here?
Jim Propp
Love this question and am not happy yet with any answers I've heard. I want a definition that does not just say what math is, but also distinguishes it from its near neighbors (e.g. Computer science and logic) and distinguishes among ways of practicing math, e.g. Pure vs. applied vs. recreational. Thoughts? Sent from my iPhone
On Jun 11, 2015, at 3:23 PM, James Propp <jamespropp@gmail.com> wrote:
A couple of weeks ago, in a gathering of mathematicians throwing out candidate definitions of mathematics, I half-seriously ventured the opinion that mathematics is the subset of philosophy consisting of those philosophical questions that actually have answers, along with the answers to those questions.
But I don't think this is original. Whom am I quoting (or paraphrasing) here?
Jim Propp _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
Mathematics is the art of formal symbolic manipulation. Applied mathematics is the art of establishing symbolic representations of real-world phenomena.
-----Original Message----- From: math-fun [mailto:math-fun- bounces+davidwwilson=comcast.net@mailman.xmission.com] On Behalf Of SCOTT KIM Sent: Friday, July 03, 2015 3:43 PM To: math-fun Subject: Re: [math-fun] Definition of mathematics
Love this question and am not happy yet with any answers I've heard. I want a definition that does not just say what math is, but also distinguishes it from its near neighbors (e.g. Computer science and logic) and distinguishes among ways of practicing math, e.g. Pure vs. applied vs. recreational. Thoughts?
Sent from my iPhone
On Jun 11, 2015, at 3:23 PM, James Propp <jamespropp@gmail.com> wrote:
A couple of weeks ago, in a gathering of mathematicians throwing out candidate definitions of mathematics, I half-seriously ventured the opinion that mathematics is the subset of philosophy consisting of those philosophical questions that actually have answers, along with the answers to those questions.
But I don't think this is original. Whom am I quoting (or paraphrasing) here?
Jim Propp _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
I think David and I are on the same wavelength. Scott's objection will apply equally well to David's formulation. I would answer Scott, at least in part, with my longtime suspicion that logic and computer science are not considered part of mathematics only through historical accident, the same sort of accident that makes dentistry not a branch of medicine. On Fri, Jul 3, 2015 at 9:56 PM, David Wilson <davidwwilson@comcast.net> wrote:
Mathematics is the art of formal symbolic manipulation. Applied mathematics is the art of establishing symbolic representations of real-world phenomena.
-----Original Message----- From: math-fun [mailto:math-fun- bounces+davidwwilson=comcast.net@mailman.xmission.com] On Behalf Of SCOTT KIM Sent: Friday, July 03, 2015 3:43 PM To: math-fun Subject: Re: [math-fun] Definition of mathematics
Love this question and am not happy yet with any answers I've heard. I want a definition that does not just say what math is, but also distinguishes it from its near neighbors (e.g. Computer science and logic) and distinguishes among ways of practicing math, e.g. Pure vs. applied vs. recreational. Thoughts?
Sent from my iPhone
On Jun 11, 2015, at 3:23 PM, James Propp <jamespropp@gmail.com> wrote:
A couple of weeks ago, in a gathering of mathematicians throwing out candidate definitions of mathematics, I half-seriously ventured the opinion that mathematics is the subset of philosophy consisting of those philosophical questions that actually have answers, along with the answers to those questions.
But I don't think this is original. Whom am I quoting (or paraphrasing) here?
Jim Propp _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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I'm not understanding this distinguishing thing. Doesn't mathematics include logic, and have a large overlap with computer science (at least finite math, and complexity theory)? —Dan
On Jul 3, 2015, at 12:43 PM, SCOTT KIM <scottekim1@gmail.com> wrote:
Love this question and am not happy yet with any answers I've heard. I want a definition that does not just say what math is, but also distinguishes it from its near neighbors (e.g. Computer science and logic) and distinguishes among ways of practicing math, e.g. Pure vs. applied vs. recreational. Thoughts?
Sent from my iPhone
On Jun 11, 2015, at 3:23 PM, James Propp <jamespropp@gmail.com> wrote:
A couple of weeks ago, in a gathering of mathematicians throwing out candidate definitions of mathematics, I half-seriously ventured the opinion that mathematics is the subset of philosophy consisting of those philosophical questions that actually have answers, along with the answers to those questions.
But I don't think this is original. Whom am I quoting (or paraphrasing) here?
Welcome (back?), Scott. It's hard to disentangle the questions "What is mathematics?", "What do mathematicians do?", and "What is a mathematician?" One of my LEAST favorite answers to the last of these questions is Lord Kelvin's: "A mathematician is one to whom *that* [the formula for the integral of exp(-x^2) as x goes from minus infinity to infinity] is as obvious as that twice two makes four is to you." I dislike that for a whole bunch of reasons, not the least of which is that it implies that I am not a mathematician --- I can't even remember whether it's pi or 2 pi inside the square root. Relating to a different point from Scott's email, here's a fun story that's probably almost completely true (I may have minor details wrong): Probabilists Robin Pemantle and Jeff Steif were at an airport together when Robin said he could often spot mathematicians by their appearance, and when pressed to demonstrate this knack, opined that a particular fellow in the waiting area looked like a mathematician. They asked him, but he said he wasn't a mathematician. Later, on the flight, Robin saw the same person reading a yellow Springer-Verlag text. "Are you -sure- you're not a mathematician?" asked Robin. "I'm not a mathematician," the stranger replied, "I'm a theoretical computer scientist!" Jim Propp On Friday, July 3, 2015, SCOTT KIM <scottekim1@gmail.com> wrote:
Love this question and am not happy yet with any answers I've heard. I want a definition that does not just say what math is, but also distinguishes it from its near neighbors (e.g. Computer science and logic) and distinguishes among ways of practicing math, e.g. Pure vs. applied vs. recreational. Thoughts?
Sent from my iPhone
On Jun 11, 2015, at 3:23 PM, James Propp <jamespropp@gmail.com <javascript:;>> wrote:
A couple of weeks ago, in a gathering of mathematicians throwing out candidate definitions of mathematics, I half-seriously ventured the opinion that mathematics is the subset of philosophy consisting of those philosophical questions that actually have answers, along with the answers to those questions.
But I don't think this is original. Whom am I quoting (or paraphrasing) here?
Jim Propp _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com <javascript:;> https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com <javascript:;> https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
That's funny and sad. I'll have to ask Robin about that the next time I see him. I've been going to the Friday lunchtime theoretical CS seminars at the Princeton Univ. for the last 20 years. For the life of me I can't see why what they're doing isn't mathematics. It's just a weird historical accident. However, I do notice a difference in the knowledge of theoretical CS people -- they usually (there are a few exceptions) have much less analysis, virtually no topology, and much less Abstract Algebra, then mathematicians. Victor On Sat, Jul 4, 2015 at 10:47 AM, James Propp <jamespropp@gmail.com> wrote:
Welcome (back?), Scott.
It's hard to disentangle the questions "What is mathematics?", "What do mathematicians do?", and "What is a mathematician?" One of my LEAST favorite answers to the last of these questions is Lord Kelvin's: "A mathematician is one to whom *that* [the formula for the integral of exp(-x^2) as x goes from minus infinity to infinity] is as obvious as that twice two makes four is to you." I dislike that for a whole bunch of reasons, not the least of which is that it implies that I am not a mathematician --- I can't even remember whether it's pi or 2 pi inside the square root.
Relating to a different point from Scott's email, here's a fun story that's probably almost completely true (I may have minor details wrong): Probabilists Robin Pemantle and Jeff Steif were at an airport together when Robin said he could often spot mathematicians by their appearance, and when pressed to demonstrate this knack, opined that a particular fellow in the waiting area looked like a mathematician. They asked him, but he said he wasn't a mathematician. Later, on the flight, Robin saw the same person reading a yellow Springer-Verlag text. "Are you -sure- you're not a mathematician?" asked Robin. "I'm not a mathematician," the stranger replied, "I'm a theoretical computer scientist!"
Jim Propp
On Friday, July 3, 2015, SCOTT KIM <scottekim1@gmail.com> wrote:
Love this question and am not happy yet with any answers I've heard. I want a definition that does not just say what math is, but also distinguishes it from its near neighbors (e.g. Computer science and logic) and distinguishes among ways of practicing math, e.g. Pure vs. applied vs. recreational. Thoughts?
Sent from my iPhone
On Jun 11, 2015, at 3:23 PM, James Propp <jamespropp@gmail.com <javascript:;>> wrote:
A couple of weeks ago, in a gathering of mathematicians throwing out candidate definitions of mathematics, I half-seriously ventured the opinion that mathematics is the subset of philosophy consisting of those philosophical questions that actually have answers, along with the answers to those questions.
But I don't think this is original. Whom am I quoting (or paraphrasing) here?
Jim Propp _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com <javascript:;> https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com <javascript:;> https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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When I talk to kids about math, I tell them that it's not just the study of numbers, but the study of forms, shapes, and patterns. I think of it as the study of structure. Cris On Jul 4, 2015, at 8:47 AM, James Propp <jamespropp@gmail.com> wrote:
Welcome (back?), Scott.
It's hard to disentangle the questions "What is mathematics?", "What do mathematicians do?", and "What is a mathematician?" One of my LEAST favorite answers to the last of these questions is Lord Kelvin's: "A mathematician is one to whom *that* [the formula for the integral of exp(-x^2) as x goes from minus infinity to infinity] is as obvious as that twice two makes four is to you." I dislike that for a whole bunch of reasons, not the least of which is that it implies that I am not a mathematician --- I can't even remember whether it's pi or 2 pi inside the square root.
Relating to a different point from Scott's email, here's a fun story that's probably almost completely true (I may have minor details wrong): Probabilists Robin Pemantle and Jeff Steif were at an airport together when Robin said he could often spot mathematicians by their appearance, and when pressed to demonstrate this knack, opined that a particular fellow in the waiting area looked like a mathematician. They asked him, but he said he wasn't a mathematician. Later, on the flight, Robin saw the same person reading a yellow Springer-Verlag text. "Are you -sure- you're not a mathematician?" asked Robin. "I'm not a mathematician," the stranger replied, "I'm a theoretical computer scientist!"
Jim Propp
On Friday, July 3, 2015, SCOTT KIM <scottekim1@gmail.com> wrote:
Love this question and am not happy yet with any answers I've heard. I want a definition that does not just say what math is, but also distinguishes it from its near neighbors (e.g. Computer science and logic) and distinguishes among ways of practicing math, e.g. Pure vs. applied vs. recreational. Thoughts?
Sent from my iPhone
On Jun 11, 2015, at 3:23 PM, James Propp <jamespropp@gmail.com <javascript:;>> wrote:
A couple of weeks ago, in a gathering of mathematicians throwing out candidate definitions of mathematics, I half-seriously ventured the opinion that mathematics is the subset of philosophy consisting of those philosophical questions that actually have answers, along with the answers to those questions.
But I don't think this is original. Whom am I quoting (or paraphrasing) here?
Jim Propp _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com <javascript:;> https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com <javascript:;> https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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I would say that mathematics is the study of the descendants of the concept of number. The arguments about how to separate the fields are something like the arguments about how to classify species, and very often you'll get mules and ligers. On Sat, Jul 4, 2015 at 9:29 AM, Cris Moore <moore@santafe.edu> wrote:
When I talk to kids about math, I tell them that it's not just the study of numbers, but the study of forms, shapes, and patterns. I think of it as the study of structure.
Cris
On Jul 4, 2015, at 8:47 AM, James Propp <jamespropp@gmail.com> wrote:
Welcome (back?), Scott.
It's hard to disentangle the questions "What is mathematics?", "What do mathematicians do?", and "What is a mathematician?" One of my LEAST favorite answers to the last of these questions is Lord Kelvin's: "A mathematician is one to whom *that* [the formula for the integral of exp(-x^2) as x goes from minus infinity to infinity] is as obvious as that twice two makes four is to you." I dislike that for a whole bunch of reasons, not the least of which is that it implies that I am not a mathematician --- I can't even remember whether it's pi or 2 pi inside the square root.
Relating to a different point from Scott's email, here's a fun story that's probably almost completely true (I may have minor details wrong): Probabilists Robin Pemantle and Jeff Steif were at an airport together when Robin said he could often spot mathematicians by their appearance, and when pressed to demonstrate this knack, opined that a particular fellow in the waiting area looked like a mathematician. They asked him, but he said he wasn't a mathematician. Later, on the flight, Robin saw the same person reading a yellow Springer-Verlag text. "Are you -sure- you're not a mathematician?" asked Robin. "I'm not a mathematician," the stranger replied, "I'm a theoretical computer scientist!"
Jim Propp
On Friday, July 3, 2015, SCOTT KIM <scottekim1@gmail.com> wrote:
Love this question and am not happy yet with any answers I've heard. I want a definition that does not just say what math is, but also distinguishes it from its near neighbors (e.g. Computer science and logic) and distinguishes among ways of practicing math, e.g. Pure vs. applied vs. recreational. Thoughts?
Sent from my iPhone
On Jun 11, 2015, at 3:23 PM, James Propp <jamespropp@gmail.com <javascript:;>> wrote:
A couple of weeks ago, in a gathering of mathematicians throwing out candidate definitions of mathematics, I half-seriously ventured the opinion that mathematics is the subset of philosophy consisting of those philosophical questions that actually have answers, along with the answers to those questions.
But I don't think this is original. Whom am I quoting (or paraphrasing) here?
Jim Propp _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com <javascript:;> https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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-- Mike Stay - metaweta@gmail.com http://www.cs.auckland.ac.nz/~mike http://reperiendi.wordpress.com
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
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> 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 https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
-- Andy.Latto@pobox.com
That definition is disturbingly time-dependent. People couldn't imagine how a continuous function could possibly be nowhere-differentiable, until Weierstrass and Blancmagne found their famous examples.
On Sat, Jul 4, 2015 at 3:30 PM, James Propp <jamespropp@gmail.com> 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 https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
-- Andy.Latto@pobox.com
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On Jul 5, 2015, at 5:23 PM, Adam P. Goucher <apgoucher@gmx.com> wrote:
until Weierstrass and Blancmagne found their famous examples. I was aware that sentient extragalactic Blancmanges posed a threat to Wimbledon. Their attack on the differentiability of functions is news to me.
-Veit
Actually, I believe the first example was Bolzano's; his construction (see for instance http://demonstrations.wolfram.com/BolzanosFunction/) should be better known. What's especially curious is that the construction can be viewed as a derandomized version of F. B. Knight's construction of Brownian motion developed more than a century later. This stochastic construction yields a continuous nowhere differentiable function with probability 1. (Gotta write that up one of these days...) Jim Propp On Monday, July 6, 2015, Veit Elser <ve10@cornell.edu <javascript:_e(%7B%7D,'cvml','ve10@cornell.edu');>> wrote:
On Jul 5, 2015, at 5:23 PM, Adam P. Goucher <apgoucher@gmx.com> wrote:
until Weierstrass and Blancmagne found their famous examples. I was aware that sentient extragalactic Blancmanges posed a threat to Wimbledon. Their attack on the differentiability of functions is news to me.
-Veit _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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 Poincaré <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:;>
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-- Thane Plambeck tplambeck@gmail.com http://counterwave.com/
I'm in agreement with Vi Hart's definition, quoted above. (Is Vi among us? She should be. I know George is.) But Vi does not go quite far enough, in specifying what *kind* of rules we mean. The rules of baseball or traffic laws don't qualify; the rules we are talking about are *formal* rules in a very specific sense. On Sun, Jul 5, 2015 at 5:29 PM, Thane Plambeck <tplambeck@gmail.com> 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 Poincaré <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-Stewart...
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-Stewart...
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
-- Thane Plambeck tplambeck@gmail.com http://counterwave.com/ _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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 Poincaré <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:;>
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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
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
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
Or the other option, advocated by Bruno Marchal, is to embrace it and say that theology is the study of what is most basic and (per Kronecker) the integers and Church-Turing computation are most basic and God is all the truths of arithmetic. Bruno regards anthropomorphic theism as corruption and politicalization of *real* theology. Brent Meeker
There's a clever proof of this that's hard to forget once you see it: Let I = Integral_{-oo,oo} exp(-x^2) dx. Then I^2 = [Integral_{-oo,oo} exp(-x^2) dx] [Integral_{-oo,oo} exp(-y^2) dy] = Integral_{the xy-plane} exp(-(x^2 + y^2)) dx dy = Integral_{0,2pi} [Integral_{0,oo} exp(-r^2) r dr] dt = 2pi Integral_{0,oo} exp(-r^2) r dr = 2pi (-1/2) Integral_{0,oo} exp(-r^2) -2r dr oo = -pi exp(-r^2)] 0 = pi So I = sqrt(pi). —Dan
[the formula for the integral of exp(-x^2) as x goes from minus infinity to infinity]
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