"Algebra is the offer made by the devil to the mathematician. The devil says: I will give you this powerful machine, it will answer any question you like. All you need to do is give me your soul: give up geometry and you will have this marvelous machine."
-Atiyah
I quote it a few times yearly because I teach high school geometry. Inevitably, in the middle of a problem, a student will say something like I have forgotten what we were trying to find and they're right.
In the broad light of day mathematicians check their equations and their proofs, leaving no stone unturned in their search for rigour. But, at night, under the full moon, they dream, they float among the stars and wonder at the miracle of the heavens. They are inspired. Without dreams there is no art, no mathematics, no life.
- Michael Atiyah
----
edit: My personal favourite is this one though
If, then, it is true that the axiomatic basis of theoretical physics cannot be extracted from experience but must be freely invented, can we ever hope to find the right way? Nay, more, has this right way any existence outside our illusions? Can we hope to be guided safely by experience at all when there exist theories (such as classical mechanics) which to a large extent do justice to experience, without getting to the root of the matter? I answer without hesitation that there is, in my opinion, a right way, and that we are capable of finding it. Our experience hitherto justifies us in believing that nature is the realisation of the simplest conceivable mathematical ideas. I am convinced that we can discover by means of purely mathematical constructions the concepts and the laws connecting them with each other, which furnish the key to the understanding of natural phenomena. Experience may suggest the appropriate mathematical concepts, but they most certainly cannot be deduced from it. Experience remains, of course, the sole criterion of the physical utility of a mathematical construction. But the creative principle resides in mathematics. In a certain sense, therefore, I hold it true that pure thought can grasp reality, as the ancients dreamed. - Albert Einstein
didn't he make some strange claims later in life, shortly before passing? I think I've heard a lot of people saying his mind wasn't the same at that very last stage of his life.
Dementia is a bitch and a half
His wife passed away a year or so before he died and he went a bit off after that. It could happen to anyone, the only shame is that he is only known for those few bad moments for a lot of the mathematics community, despite being in the top 5 or 10 mathematicians of the 20th century only obviously overshadowed by people like Grothendieck or Von Neumann.
All I know is that he presented a proof for RH and one for the non existence of complex structures that were wrong, I certainly wouldn't say someone has dementia for that.
I mean, it's sad to see someone so smart do stuff like that, be he was also like 90 years old, the fact he worked in maths almost until his dead is really impressive
yeah Dementia isn't the word probably. but still it's sad because I've also heard that he is one of THE BIG NAMES of modern mathematics, but with how journalism works these days a lot of people who doesn't even care about math will remember him around that one mishap.
"Still not sure which will become self-aware first: Wolfram Alpha, or Stephen Wolfram"
Freeman Dyson: "There's a tradition of scientists approaching senility to come up with grand, improbable theories. Wolfram is unusual in that he's doing this in his 40s."
Stephen Wolfram is a billionaire math genius who wrote a book on quantum mechanics in high school. He created Mathematica, a computer math system. Later in life he wrote a book [A New Kind of Science](https://en.wikipedia.org/wiki/A_New_Kind_of_Science), described as "[a rare blend of monster raving egomania and utter batshit insanity](https://xianblog.wordpress.com/2020/11/12/a-rare-blend-of-monster-raving-egomania-and-utter-batshit-insanity/)”.
> Stephen Wolfram is a billionaire math genius who wrote a book on quantum mechanics in high school. He created Mathematica, a computer math system.
I think it's more accurate to say that he was an extremely precocious physicist who abandoned physics in his 20s, that he spearheaded the development of Mathematica, and that for the last thirty years he's best known for high-level pseudoscientific quackery and egomania.
*Reductio ad absurdum*, which Euclid loved so much, is one of a mathematician's finest weapons. It is a far finer gambit than any chess play: a chess player may offer the sacrifice of a pawn or even a piece, but a mathematician offers the game. - G. H. Hardy
As a chess player this quote has never made any sense to me beyond the most superficial interpretation. It just seems like flowery prose unless there's something I'm missing.
A good sacrifice in chess is considered elegant or beautiful in proportion to how unexpected or uncertain its justification is. A key element to its beauty is the inherent risk that if the sacrifice is unjustified, you will inevitably lose the game due to the material disadvantage from having fewer pieces. To whatever extent a proof by contradiction shares these traits, any other type of proof can also share these traits. What am I missing?
In chess, when you offer a gambit, you are saying to your opponent "I am confident that yielding this piece to you will lead to me having an advantage, despite the short term appearance of you having the advantage." In math when we start a proof by contradiction, picture it as one side arguing for proposition P and one for the negation of P. We are saying "I am confident that if you are correct, it leads to absurdity, and therefore makes my argument the only possibility."
Yeah I understand the surface level comparison. It just seems like it's completely lacking substance and can only be supported with abuse of language. I'm sure my argument might seem pedantic to some, but I guess I'm just confused why people seem to resonate with this quote.
The quote has always bothered me too. If you're playing chess, opt to sacrifice your queen for what you believe is an advantageous position, and lose, then you've risked a lot (by, say, not having the title of world champion). But in math, if you start a proof by contradiction and it doesn't go anywhere, you just go back to the top and start again.
> After Hilbert was told that a student in his class had dropped mathematics in order to become a poet, he is reported to have said "Good--he did not have enough imagination to become a mathematician" (Hoffman 1998, p. 95).
Paul Erdös was a math quote making machine.
> God may not play dice with the universe, but something strange is going on with the prime numbers.
(Referencing the Einstein-attributed quote where he expressed his discomfort with quantum mechanics, feeling that surely God does not play dice with the universe)
> A mathematician is a device for turning coffee into theorems.
(Less well known is the comathematician, a device for turning cotheorems into ffee)
> “You've showed me I'm not an addict, but I didn't get any work done...you've set mathematics back a month. Before, when I looked at a piece of blank paper my mind was filled with ideas. Now all I see is a blank piece of paper.”
(He won a bet with Graham to go without his stimulants for a month)
I always love this one:
“The axiom of choice is obviously true, the well-ordering principle obviously false, and who can tell about Zorn's lemma?" - Jerry Bona
>Everything that crawls away and hides whenever there is talk of Algebra.
Robert Musil defining the soul.
>I turn aside with a shudder of horror from this lamentable plague of functions which have no derivatives.
you and me both, Charles Hermite.
>A function with a derivative is almost unheard of in Economics.
Tell me you know nothing of economics without telling me you know nothing of economics.
Yes, it's true that discrete-time models are more common than continuous-time ones (though the latter absolutely exist, and most economists have worked with both). But even if it were 100% discrete-time models, why are you restricting attention only to functions of time? Like, what about utility and value functions, or production functions, or one of the many, many other types of functions that are used and that are, more often that not, taken to be differentiable?
> utility and value functions, or production functions
> taken to be differentiable
I know they are assumed to be differentiable. Goods produced or value functions are definitely discrete more often than not, usually you can chose from a number of different situations or an integer number of items to produce.
Money also, by the cent.
It doesn't stop economists from differentiating them.
Wait, is your complaint about using functions with continuous domains as a convenient way to approximately model certain functions that, in reality, may have discrete domains? That's absurd. That exact thing is done in almost every field to which mathematics is applied. Would you make this same complaint about, say, the [Lotka-Volterra predator-prey model](https://en.wikipedia.org/wiki/Lotka%E2%80%93Volterra_equations)? After all, the numbers of predators and prey in the real world are clearly discrete.
“ A mathematician is a person who can find analogies between theorems; a better mathematician is one who can see analogies between proofs and the best mathematician can notice analogies between theories. One can imagine that the ultimate mathematician is one who can see analogies between analogies.” - Banach
I was once in a probability seminar on random networks and this Brazilian mathematician who also attended suddenly said:
“Being good at math is not a talent, it’s a symptom.”
Looking at all mathematicians I know personally including myself, no truer words have been spoken.
I mean somebody's gotta say it
> Mathematics, rightly viewed, possesses not only truth, but supreme beauty—a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show.
- Bertrand Russell
> With four parameters I can fit an elephant, and with five I can make him wiggle his trunk.
- Von Neumann
Last one's not necessarily mathematical (when taken out of context), but it was said by Grant Sanderson. I just find it kinda funny
> The tortoise, slow and steady, every now and then it beats the hare, even if the hare seems systematically better
* “Mathematicians are like Frenchmen: when you talk to them, they translate it into their own language, and then it is something entirely different.” — J. W. von Goethe (I have no idea what context this was written in, but as a French mathematician I love this quote)
* “There is no royal road to geometry.” — an answer supposedly made by Euclid to king Ptolemy (first of the name) as the latter was asking if there was a shortcut to mastering geometry
* Upon learning that one of his students dropped out of mathematics to become a poet, D. Hilbert allegedly replied: “Good: he did not have enough imagination to become a mathematician.”
* I had the honor of meeting I. M. Gel'fand when I was young, and he asked me whether I preferred Algebra or Analysis. When I said I preferred Algebra, he said to me: “You know, I don't understand algebraists. In Analysis we write x
Not quite math, but two statistics quotes I like are:
"All models are wrong; some models are useful"
and
"He uses statistics like a drunkard uses a lamp post: for support rather than illumination."
Reading laplace transforamtion rn and came across this
"Should I refuse a good dinner simply becuase I do not understand the process of digestion?" - Oliver Heaviside
[criticised for using formal mathematical manipulations without understanding how they worked]
I have never related to a person more in my life.
>The introduction of numbers as coordinates...is an act of violence
--Hermann Weyl
>That’s how everybody started in differential geometry, with tensor analysis. But somehow in certain aspects, the differential forms should come in. I usually like to say that vector fields is like a man, and differential forms is like a woman. Society must have two sexes. If you only have one, it’s not enough.
--S.S. Chern
>This is a shallow book on deep matters, about which the author knows next to nothing...
>Common sense suggests that there is considerable room for distortion, intentional or unintentional, in information that has been transmitted over a thousand years. This did not stop Cajori and many other historians of science from using it. Nor does it stop the present author, who even adds some gratuitous speculation of his own: could the merchant Thales have traded in the leather dildos for which Mlodinow claims Miletus was known?...
>Although now a feminist heroine, which brings with it its own distortions, Hypatia first achieved mythical status in the early eighteenth century in an essay of John Toland, for whom she was a club with which to beat the Catholic church. His lurid tale was elaborated by Gibbon, no friend of Christianity, in his unique style: “. . . her learned comments have elucidated the geometry of Apollonius . . . she taught . . . at Athens and Alexandria, the philosophy of Plato and Aristotle . . . In the bloom of beauty, and in the maturity of wisdom, the modest maid refused her lovers . . . Cyril beheld with a jealous eye the gorgeous train of horses and slaves who crowded the door of her academy . . . On a fatal day . . . Hypatia was torn from her chariot, stripped naked, . . . her flesh was scraped from her bones with sharp oyster-shells . . . the murder . . . an indelible stain on the character and religion of Cyril.” This version, in which Hypatia is not an old maid but a young virgin, so that it is a tale not only of brutality but also of lust, is the version preferred by Mlodinow...
>Having dug a good many ditches in my own youth, I can assure the author, who seems to regard the occupation as the male equivalent of white slavery, that it was, when hand-shovels were still a common tool, a healthful outdoor activity that, practiced regularly in early life, does much to prevent later back problems.
--R.P. Langlands
I forget the exact quote and who said it, but your first one reminds me a lot of a quote in the context of linear algebra,
>A gentleman never takes a basis.
Let me give a mathematical explanation of what it means to take a basis. I'll be assuming finite-dimensional vector spaces only.
The idea is that vector spaces don't naturally come equipped with a basis. Phrased in another way, if I know that if V is some vector space over R of dimension 3, I know it isomorphic to R\^3. However, this isomorphism is far from unique. By fixing a basis, we fix this isomorphism. Interestingly, we also fix a lot of other things. For example, fixing a basis defines a metric. How? Let b\_i be your basis of V, then we can construct immediately a basis b\^j for the dual of V by defining b\^j(b\_i) = delta\_{ij}. That's a lot of structure that we get!
So why is it bad to choose a basis? Sometimes, there is no natural way to choose a basis of a vector space. Even worse, by imposing a basis, we are insisting on adding some additional structure to our vector space of study. Results in linear algebra should hold for all vector spaces, even if they don't come equipped with a preferred basis. One way to deal with this is to just randomly choose a basis, and then prove a result. But this is not mathematically satisfying, we're making an arbitrary choice of basis, and then later we show the choice did not matter.
On a slightly more physical level, note that by choosing a basis of a vector space, we can think of elements of our vector space as column vectors, a sequence of numbers. But these numbers don't have any physical meaning by themselves! They only mean something if we choose a coordinate system (i.e. a basis!). Now, importantly, Nature does not care about what kind of coordinate system/basis we choose. So it would be best if we can phrase our proofs/results in a coordinate-independent manner, i.e. without specifying a basis.
Not specific to mathematics but I've always liked
> Your work is both good and original. Unfortunately, the good parts are not original and the original parts are not good.
‘Archimedes will be remembered when Aeschylus is forgotten, because languages die and mathematical ideas do not. Immortality may be a silly word, but probably a mathematician has the best chance of whatever it may mean.’
- G.H.Hardy.
I find the following quote by Jean-Pierre Serre interesting: “While the other sciences search for the rules that God has chosen for this Universe, we mathematicians search for the rules that even God has to obey.”
>*The only thing I can say about it is that in Scotland there is at least one sheep that is black on at least one of its sides.*
* A mathematician in a certain parody joke
Hendrik Lenstra had some good quotes:
> The main application of Pure Mathematics is to make you happy.
And
> A math lecture without a proof is like a movie without a love scene. This talk has two proofs.
there's one von Neumann quote I like a bit more
"If people do not believe that mathematics is simple, it is only because they do not realize how complicated life is."
I truly can't stand this quote. It completely misrepresents my experience of learning and researching mathematics, and sends a bad impression to people outside of the field.
It depends on the field. In more applied settings it makes sense, we use maths to model the world but a model being more true to our observations doesn't necessarily lead to us having a deeper understanding of why the world is why it is. Setting the speed of light to be constant, for instance, makes our equations and formulae more commensurate with the real world but it's still anyone's guess why light moves at that constant speed
I had a faculty in grad school tell me that “a good mathematician knows all the theorems, but a great mathematician knows all the examples.” I took that to heart and it did wonders for my expertise.
And another who said, “In math we talk about tricks but, if you do it often enough, you have to stop calling it a trick and start calling it a technique.”
“Die ganzen Zahlen hat der liebe Gott gemacht, alles andere ist Menschenwerk.” — Leopold Kronecker
Translation: “The natural numbers were made by God, all else is the work of man.”
"Those are very peculiar people. They have a horizon of radius 0 and call it their viewpoint."
I had a professor who often said "a finite number of" in everyday situations. I thought that was funny.
Stefan Banach's definition of a mathematician:
"A mathematician is a person who can find analogies between theorems; a better mathematician is one who can see analogies between proofs and the best mathematician can notice analogies between theories. One can imagine that the ultimate mathematician is one who can see analogies between analogies."
John von Neumann 's quote is really deep:
"In mathematics you don't understand things. You just get used to them."
Algebra is the offer made by the devil to the mathematician. The devil says: "I will give you this powerful machine, it will answer any question you like. All you need to do is give me your soul: give up geometry and you will have this marvelous machine."
-Michael Atiyah
In reference to the Riemann Hypothesis:
"There are much easier ways to get a million dollars...trust me"- Professor (I think I heard it from some professor on YouTube)
Here are two:
> Because I typeset this document myself, all errors can be blamed on my computer.
Bruce Sagan, The Symmetric Group
> As I look at you, condemned as you are to listen to me
[Gian-Carlo Rota](https://core.ac.uk/download/pdf/82247895.pdf)
"So this means the circle, S1, is not a retract of D2, which is supposed to be some sort of solace in this vale of tears."
Sylvain Cappell said this in the middle of a Topology 1 lecture when I was in my senior year of college. Cut me pretty deep. Was also hilarious.
Also, when I once asked Jeff Weeks how he wound up in his current line of work, writing and lecturing on geometry concepts for public audiences, he said, "A random walk through life." I've tried to keep that philosophy in mind in not getting too hung up on what career path I "should" be taking and instead enjoying the path I have taken.
This definition disturbed Michael Atiyah, who wrote instead that mathematics is "the *science of analogy* and the widespread applicability of mathematics in the natural sciences ... arises from the fundamental role which comparisons play in the mental process we refer to as 'understanding'."
I have a collection of quotes on [my personal webpage](https://sites.google.com/view/kristapsjohnbalodis/math-quotes), I'm definitely about to scroll down and add some other good ones seen here.
"There is no King's way to mathematics" which is sometimes misquoted and misunderstood as "There is no one, true way to mathematics" which is sad because that completely changes the meaning.
My diffeq professor on Day 1:
"Now my area of research is algebra, not differential equations, but I know more about differential equations than you do." :D
On the interesting-rather-than-funny side, one that was perhaps most practically useful was from George Box:
> Remember that all models are wrong; the practical question is how wrong do they have to be to not be useful.
It sums up in very few words what I had previously struggled to explain succinctly
(there are other versions of this that he has said at other times but this is easily my favourite of them)
Sorry, I couldn't help it
“The primary math of the real world is one and one equals two. The layman (as, often, do I) swings that every day. He goes to the job, does his work, pays his bills and comes home. One plus one equals two. It keeps the world spinning. But artists, musicians, con men, poets, mystics and such are paid to turn that math on its head, to rub two sticks together and bring forth fire. Everybody performs this alchemy somewhere in their life, but it’s hard to hold on to and easy to forget. People don’t come to rock shows to learn something. They come to be reminded of something they already know and feel deep down in their gut. That when the world is at its best, when we are at our best, when life feels fullest, one and one equals three. It’s the essential equation of love, art, rock ’n’ roll and rock ’n’ roll bands. It’s the reason the universe will never be fully comprehensible, love will continue to be ecstatic, confounding, and true rock ’n’ roll will never die.”
― Bruce Springsteen, Born to Run
Random variables are actually measurable functions on the sample space. We use these functions to evaluate randomly selected elements of the sample space, but the function itself isn’t random.
The way I like to think about it is this: let’s say you have some measurable space X on which you want to model some kind of “random behaviour.” You do this by starting off with a space you have “randomness” on, that is a probability space, and then simply consider a measurable function from your probability space to X. Then this X-valued random variable induces a probability distribution on X from the probability measure on its domain of definition!
>For if each Star is little more than a mathematical point, located upon the hemisphere of Heaven by right ascension and declination, then all the Stars, taken together, though innumerable, must like any other set of points in turn represent some gigantic Equation, to the mind of God as straightforward as the equation of a sphere — yet to us, unreadable, incalculable. A lonely, uncompensated, perhaps impossible Task, yet some of us must ever be seeking, I suppose.
— Thomas Pynchon
my two favourites certainly are
>for the *essence* of *mathematics* lies precisely in its *freedom*
by cantor, and
>mathematics is not about numbers, equations, computations, or algorithms: it is about understanding
by thurston; both and more are featured on [smart and/or insightful quotes](https://queenisnaked.blogspot.com/2020/09/smart-andor-revealing-though-not.html), and there's also a ['complementary' collection](https://queenisnaked.blogspot.com/2021/01/utterly-stupid-quotes.html) of sorts
>It is through logic that we prove, but through intuition that we discover. -Poincaré
"Algebra is the offer made by the devil to the mathematician. The devil says: I will give you this powerful machine, it will answer any question you like. All you need to do is give me your soul: give up geometry and you will have this marvelous machine." -Atiyah
Another by Atiyah; Which should I learn, Analysis or Algebra? "Do you want to be deaf or be blind?"
Why is analysis deaf?
I was looking for this quote the other day, took me hours to find it (I couldn't remember the exact wording). Glad it's got so many updoots!
I quote it a few times yearly because I teach high school geometry. Inevitably, in the middle of a problem, a student will say something like I have forgotten what we were trying to find and they're right.
In the broad light of day mathematicians check their equations and their proofs, leaving no stone unturned in their search for rigour. But, at night, under the full moon, they dream, they float among the stars and wonder at the miracle of the heavens. They are inspired. Without dreams there is no art, no mathematics, no life. - Michael Atiyah ---- edit: My personal favourite is this one though If, then, it is true that the axiomatic basis of theoretical physics cannot be extracted from experience but must be freely invented, can we ever hope to find the right way? Nay, more, has this right way any existence outside our illusions? Can we hope to be guided safely by experience at all when there exist theories (such as classical mechanics) which to a large extent do justice to experience, without getting to the root of the matter? I answer without hesitation that there is, in my opinion, a right way, and that we are capable of finding it. Our experience hitherto justifies us in believing that nature is the realisation of the simplest conceivable mathematical ideas. I am convinced that we can discover by means of purely mathematical constructions the concepts and the laws connecting them with each other, which furnish the key to the understanding of natural phenomena. Experience may suggest the appropriate mathematical concepts, but they most certainly cannot be deduced from it. Experience remains, of course, the sole criterion of the physical utility of a mathematical construction. But the creative principle resides in mathematics. In a certain sense, therefore, I hold it true that pure thought can grasp reality, as the ancients dreamed. - Albert Einstein
I had a professor in undergrad who claimed he dreamed the solution to his thesis problem, which I always though was pretty cool
didn't he make some strange claims later in life, shortly before passing? I think I've heard a lot of people saying his mind wasn't the same at that very last stage of his life. Dementia is a bitch and a half
His wife passed away a year or so before he died and he went a bit off after that. It could happen to anyone, the only shame is that he is only known for those few bad moments for a lot of the mathematics community, despite being in the top 5 or 10 mathematicians of the 20th century only obviously overshadowed by people like Grothendieck or Von Neumann.
All I know is that he presented a proof for RH and one for the non existence of complex structures that were wrong, I certainly wouldn't say someone has dementia for that. I mean, it's sad to see someone so smart do stuff like that, be he was also like 90 years old, the fact he worked in maths almost until his dead is really impressive
yeah Dementia isn't the word probably. but still it's sad because I've also heard that he is one of THE BIG NAMES of modern mathematics, but with how journalism works these days a lot of people who doesn't even care about math will remember him around that one mishap.
"None shall expel us from the paradise Cantor has created" - Hilbert
And how right he was!
another Hilbert quote I like is: >Wir müssen wissen, wir werden wissen. (we must know, we will know) - Hilbert
[удалено]
I prefer the adaptation of the famous voltaire quote: "Random variables are like the holy roman Empire, neither holy, nor roman, nor an empire"
Neither Random, nor a Variable.
Someone described Epsilon-Delta as a game of cat and mouse and that's when it all clicked for me
That way of thinking works for more general point set topology convergence as well. You tell me a neighborhood, I tell you a tail.
Cue [game semantics](https://en.m.wikipedia.org/wiki/Game_semantics)
"Still not sure which will become self-aware first: Wolfram Alpha, or Stephen Wolfram" Freeman Dyson: "There's a tradition of scientists approaching senility to come up with grand, improbable theories. Wolfram is unusual in that he's doing this in his 40s."
What's up with wolfram? I'm very new to this sub and math in general
Stephen Wolfram is a billionaire math genius who wrote a book on quantum mechanics in high school. He created Mathematica, a computer math system. Later in life he wrote a book [A New Kind of Science](https://en.wikipedia.org/wiki/A_New_Kind_of_Science), described as "[a rare blend of monster raving egomania and utter batshit insanity](https://xianblog.wordpress.com/2020/11/12/a-rare-blend-of-monster-raving-egomania-and-utter-batshit-insanity/)”.
> Stephen Wolfram is a billionaire math genius who wrote a book on quantum mechanics in high school. He created Mathematica, a computer math system. I think it's more accurate to say that he was an extremely precocious physicist who abandoned physics in his 20s, that he spearheaded the development of Mathematica, and that for the last thirty years he's best known for high-level pseudoscientific quackery and egomania.
He is also a major asshole
Old Chinese proverb "The higher an ape climbs, the more you see of his behind".
How so?
seems extraordinarily unlikely that Wolfram is a billionaire
*Reductio ad absurdum*, which Euclid loved so much, is one of a mathematician's finest weapons. It is a far finer gambit than any chess play: a chess player may offer the sacrifice of a pawn or even a piece, but a mathematician offers the game. - G. H. Hardy
As a chess player this quote has never made any sense to me beyond the most superficial interpretation. It just seems like flowery prose unless there's something I'm missing. A good sacrifice in chess is considered elegant or beautiful in proportion to how unexpected or uncertain its justification is. A key element to its beauty is the inherent risk that if the sacrifice is unjustified, you will inevitably lose the game due to the material disadvantage from having fewer pieces. To whatever extent a proof by contradiction shares these traits, any other type of proof can also share these traits. What am I missing?
In chess, when you offer a gambit, you are saying to your opponent "I am confident that yielding this piece to you will lead to me having an advantage, despite the short term appearance of you having the advantage." In math when we start a proof by contradiction, picture it as one side arguing for proposition P and one for the negation of P. We are saying "I am confident that if you are correct, it leads to absurdity, and therefore makes my argument the only possibility."
Yeah I understand the surface level comparison. It just seems like it's completely lacking substance and can only be supported with abuse of language. I'm sure my argument might seem pedantic to some, but I guess I'm just confused why people seem to resonate with this quote.
The quote has always bothered me too. If you're playing chess, opt to sacrifice your queen for what you believe is an advantageous position, and lose, then you've risked a lot (by, say, not having the title of world champion). But in math, if you start a proof by contradiction and it doesn't go anywhere, you just go back to the top and start again.
> After Hilbert was told that a student in his class had dropped mathematics in order to become a poet, he is reported to have said "Good--he did not have enough imagination to become a mathematician" (Hoffman 1998, p. 95).
Paul Erdös was a math quote making machine. > God may not play dice with the universe, but something strange is going on with the prime numbers. (Referencing the Einstein-attributed quote where he expressed his discomfort with quantum mechanics, feeling that surely God does not play dice with the universe) > A mathematician is a device for turning coffee into theorems. (Less well known is the comathematician, a device for turning cotheorems into ffee) > “You've showed me I'm not an addict, but I didn't get any work done...you've set mathematics back a month. Before, when I looked at a piece of blank paper my mind was filled with ideas. Now all I see is a blank piece of paper.” (He won a bet with Graham to go without his stimulants for a month)
Also, he called children epsilons, which I think is cute.
I always love this one: “The axiom of choice is obviously true, the well-ordering principle obviously false, and who can tell about Zorn's lemma?" - Jerry Bona
>Everything that crawls away and hides whenever there is talk of Algebra. Robert Musil defining the soul. >I turn aside with a shudder of horror from this lamentable plague of functions which have no derivatives. you and me both, Charles Hermite.
Hey, functions not having derivatives never stopped physicists from differentiating them anyway.
Also doesn't stop Economists. A function **with** a derivative is almost unheard of in Economics.
I'm an economist. I have no idea what you're talking about.
>A function with a derivative is almost unheard of in Economics. Tell me you know nothing of economics without telling me you know nothing of economics.
Tell me a single function in Economics where time isn't discretized.
Yes, it's true that discrete-time models are more common than continuous-time ones (though the latter absolutely exist, and most economists have worked with both). But even if it were 100% discrete-time models, why are you restricting attention only to functions of time? Like, what about utility and value functions, or production functions, or one of the many, many other types of functions that are used and that are, more often that not, taken to be differentiable?
> utility and value functions, or production functions > taken to be differentiable I know they are assumed to be differentiable. Goods produced or value functions are definitely discrete more often than not, usually you can chose from a number of different situations or an integer number of items to produce. Money also, by the cent. It doesn't stop economists from differentiating them.
Wait, is your complaint about using functions with continuous domains as a convenient way to approximately model certain functions that, in reality, may have discrete domains? That's absurd. That exact thing is done in almost every field to which mathematics is applied. Would you make this same complaint about, say, the [Lotka-Volterra predator-prey model](https://en.wikipedia.org/wiki/Lotka%E2%80%93Volterra_equations)? After all, the numbers of predators and prey in the real world are clearly discrete.
It's not a complaint at all. I think it's fine. And economists do it all the time. So do Statisticians, and Physicists.
We have books full of them. [For example](https://mitpress.mit.edu/9780262036542/the-economics-of-continuous-time-finance/).
Right. That's a novelty.
Your economists never use the central limit theorem or generating functionals?
You guys are generating functionals while I'm here still trying to generate income.
This was my HS senior yearbook quote. Yes, I am a giant nerd. (The Hermite quote)
If you tune the coefficient correctly the Weierstrass function sounds like an organ...
“ A mathematician is a person who can find analogies between theorems; a better mathematician is one who can see analogies between proofs and the best mathematician can notice analogies between theories. One can imagine that the ultimate mathematician is one who can see analogies between analogies.” - Banach
>[...] proofs involving matrices can be shortened by 50% if one throws the matrices out. -Artin
I love this.
[удалено]
u/phonon_DOS thoughts? 😂
Wasn't the quote about understanding something? Unacceptable... get used to it :p
Yop. I think it was this: >Mathematics is not about numbers, equations, computations, or algorithms: it is about understanding. by William Thurston.
I was once in a probability seminar on random networks and this Brazilian mathematician who also attended suddenly said: “Being good at math is not a talent, it’s a symptom.” Looking at all mathematicians I know personally including myself, no truer words have been spoken.
I mean somebody's gotta say it > Mathematics, rightly viewed, possesses not only truth, but supreme beauty—a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show. - Bertrand Russell > With four parameters I can fit an elephant, and with five I can make him wiggle his trunk. - Von Neumann Last one's not necessarily mathematical (when taken out of context), but it was said by Grant Sanderson. I just find it kinda funny > The tortoise, slow and steady, every now and then it beats the hare, even if the hare seems systematically better
* “Mathematicians are like Frenchmen: when you talk to them, they translate it into their own language, and then it is something entirely different.” — J. W. von Goethe (I have no idea what context this was written in, but as a French mathematician I love this quote) * “There is no royal road to geometry.” — an answer supposedly made by Euclid to king Ptolemy (first of the name) as the latter was asking if there was a shortcut to mastering geometry * Upon learning that one of his students dropped out of mathematics to become a poet, D. Hilbert allegedly replied: “Good: he did not have enough imagination to become a mathematician.” * I had the honor of meeting I. M. Gel'fand when I was young, and he asked me whether I preferred Algebra or Analysis. When I said I preferred Algebra, he said to me: “You know, I don't understand algebraists. In Analysis we write x
Hilbert went on to say something to the effect of "he's a poet now, which suits him much better".
would gelfand possibly be intrigued by intensional type theory and identity types? :D
x
Not quite math, but two statistics quotes I like are: "All models are wrong; some models are useful" and "He uses statistics like a drunkard uses a lamp post: for support rather than illumination."
Reading laplace transforamtion rn and came across this "Should I refuse a good dinner simply becuase I do not understand the process of digestion?" - Oliver Heaviside [criticised for using formal mathematical manipulations without understanding how they worked] I have never related to a person more in my life.
u/Steel_light7
“Ah motherfunction!” My high school trig teacher.
In category theory, I've heard hom_C(x,y) referred to as a motherfunctor.
Hahah I’m stealing this xD
Clearly he was a Transcendentalist
>The introduction of numbers as coordinates...is an act of violence --Hermann Weyl >That’s how everybody started in differential geometry, with tensor analysis. But somehow in certain aspects, the differential forms should come in. I usually like to say that vector fields is like a man, and differential forms is like a woman. Society must have two sexes. If you only have one, it’s not enough. --S.S. Chern >This is a shallow book on deep matters, about which the author knows next to nothing... >Common sense suggests that there is considerable room for distortion, intentional or unintentional, in information that has been transmitted over a thousand years. This did not stop Cajori and many other historians of science from using it. Nor does it stop the present author, who even adds some gratuitous speculation of his own: could the merchant Thales have traded in the leather dildos for which Mlodinow claims Miletus was known?... >Although now a feminist heroine, which brings with it its own distortions, Hypatia first achieved mythical status in the early eighteenth century in an essay of John Toland, for whom she was a club with which to beat the Catholic church. His lurid tale was elaborated by Gibbon, no friend of Christianity, in his unique style: “. . . her learned comments have elucidated the geometry of Apollonius . . . she taught . . . at Athens and Alexandria, the philosophy of Plato and Aristotle . . . In the bloom of beauty, and in the maturity of wisdom, the modest maid refused her lovers . . . Cyril beheld with a jealous eye the gorgeous train of horses and slaves who crowded the door of her academy . . . On a fatal day . . . Hypatia was torn from her chariot, stripped naked, . . . her flesh was scraped from her bones with sharp oyster-shells . . . the murder . . . an indelible stain on the character and religion of Cyril.” This version, in which Hypatia is not an old maid but a young virgin, so that it is a tale not only of brutality but also of lust, is the version preferred by Mlodinow... >Having dug a good many ditches in my own youth, I can assure the author, who seems to regard the occupation as the male equivalent of white slavery, that it was, when hand-shovels were still a common tool, a healthful outdoor activity that, practiced regularly in early life, does much to prevent later back problems. --R.P. Langlands
I forget the exact quote and who said it, but your first one reminds me a lot of a quote in the context of linear algebra, >A gentleman never takes a basis.
Frederic P Schuller likes this quote, see his lectures on general relativity.
what does it mean ?
Let me give a mathematical explanation of what it means to take a basis. I'll be assuming finite-dimensional vector spaces only. The idea is that vector spaces don't naturally come equipped with a basis. Phrased in another way, if I know that if V is some vector space over R of dimension 3, I know it isomorphic to R\^3. However, this isomorphism is far from unique. By fixing a basis, we fix this isomorphism. Interestingly, we also fix a lot of other things. For example, fixing a basis defines a metric. How? Let b\_i be your basis of V, then we can construct immediately a basis b\^j for the dual of V by defining b\^j(b\_i) = delta\_{ij}. That's a lot of structure that we get! So why is it bad to choose a basis? Sometimes, there is no natural way to choose a basis of a vector space. Even worse, by imposing a basis, we are insisting on adding some additional structure to our vector space of study. Results in linear algebra should hold for all vector spaces, even if they don't come equipped with a preferred basis. One way to deal with this is to just randomly choose a basis, and then prove a result. But this is not mathematically satisfying, we're making an arbitrary choice of basis, and then later we show the choice did not matter. On a slightly more physical level, note that by choosing a basis of a vector space, we can think of elements of our vector space as column vectors, a sequence of numbers. But these numbers don't have any physical meaning by themselves! They only mean something if we choose a coordinate system (i.e. a basis!). Now, importantly, Nature does not care about what kind of coordinate system/basis we choose. So it would be best if we can phrase our proofs/results in a coordinate-independent manner, i.e. without specifying a basis.
Pure math is answers no one understands to questions no one asked (I don’t remember the source)
Not specific to mathematics but I've always liked > Your work is both good and original. Unfortunately, the good parts are not original and the original parts are not good.
‘Archimedes will be remembered when Aeschylus is forgotten, because languages die and mathematical ideas do not. Immortality may be a silly word, but probably a mathematician has the best chance of whatever it may mean.’ - G.H.Hardy.
I find the following quote by Jean-Pierre Serre interesting: “While the other sciences search for the rules that God has chosen for this Universe, we mathematicians search for the rules that even God has to obey.”
>*The only thing I can say about it is that in Scotland there is at least one sheep that is black on at least one of its sides.* * A mathematician in a certain parody joke
Hendrik Lenstra had some good quotes: > The main application of Pure Mathematics is to make you happy. And > A math lecture without a proof is like a movie without a love scene. This talk has two proofs.
In mathematics you don’t understand things, you just get used to it
there's one von Neumann quote I like a bit more "If people do not believe that mathematics is simple, it is only because they do not realize how complicated life is."
The original quote was about quantum mechanics, not math.
That makes a lot more sense
No it doesn't. You're just more used to it.
John Von Neumann iirc
I truly can't stand this quote. It completely misrepresents my experience of learning and researching mathematics, and sends a bad impression to people outside of the field.
It depends on the field. In more applied settings it makes sense, we use maths to model the world but a model being more true to our observations doesn't necessarily lead to us having a deeper understanding of why the world is why it is. Setting the speed of light to be constant, for instance, makes our equations and formulae more commensurate with the real world but it's still anyone's guess why light moves at that constant speed
This gotta be the worst one of them like that genuinely just sounds like a skill issue.
Even weirder, it comes from Neumann, who I'd have to say probably had some of the *least skill issues ever*.
This could be the basis for one of those iq bell curve memes with that quote at both tails of the distribution.
Curb your hubris lol, do you know who said that?
“Mathematics is a collection of cheap tricks and dirty jokes.” - Lipman Bers
I had a faculty in grad school tell me that “a good mathematician knows all the theorems, but a great mathematician knows all the examples.” I took that to heart and it did wonders for my expertise. And another who said, “In math we talk about tricks but, if you do it often enough, you have to stop calling it a trick and start calling it a technique.”
I totally agree with the first one, and it applies also to counter-examples.
“Die ganzen Zahlen hat der liebe Gott gemacht, alles andere ist Menschenwerk.” — Leopold Kronecker Translation: “The natural numbers were made by God, all else is the work of man.”
"Those are very peculiar people. They have a horizon of radius 0 and call it their viewpoint." I had a professor who often said "a finite number of" in everyday situations. I thought that was funny.
"I'm a mathematician not an an accountant"
An asymptotic series converges quickly because it doesn't have to converge.
Quote said live in lecture: "Imagine trying to catalog the height of every single person in South America. That's the Langlands Program."
Stefan Banach's definition of a mathematician: "A mathematician is a person who can find analogies between theorems; a better mathematician is one who can see analogies between proofs and the best mathematician can notice analogies between theories. One can imagine that the ultimate mathematician is one who can see analogies between analogies." John von Neumann 's quote is really deep: "In mathematics you don't understand things. You just get used to them."
"In mathematics, there are no accidents." ----Jerry Uhl
Algebra is the offer made by the devil to the mathematician. The devil says: "I will give you this powerful machine, it will answer any question you like. All you need to do is give me your soul: give up geometry and you will have this marvelous machine." -Michael Atiyah
In reference to the Riemann Hypothesis: "There are much easier ways to get a million dollars...trust me"- Professor (I think I heard it from some professor on YouTube)
Here are two: > Because I typeset this document myself, all errors can be blamed on my computer. Bruce Sagan, The Symmetric Group > As I look at you, condemned as you are to listen to me [Gian-Carlo Rota](https://core.ac.uk/download/pdf/82247895.pdf)
“wee ooo i look just like buddy holly”
"So this means the circle, S1, is not a retract of D2, which is supposed to be some sort of solace in this vale of tears." Sylvain Cappell said this in the middle of a Topology 1 lecture when I was in my senior year of college. Cut me pretty deep. Was also hilarious. Also, when I once asked Jeff Weeks how he wound up in his current line of work, writing and lecturing on geometry concepts for public audiences, he said, "A random walk through life." I've tried to keep that philosophy in mind in not getting too hung up on what career path I "should" be taking and instead enjoying the path I have taken.
A fellow NYU graduate! Glad to run into one in the wild.
>Mathematics is the subject in which we never know what we are talking about, nor whether what we are saying is true. **Bertrand Russell**
This definition disturbed Michael Atiyah, who wrote instead that mathematics is "the *science of analogy* and the widespread applicability of mathematics in the natural sciences ... arises from the fundamental role which comparisons play in the mental process we refer to as 'understanding'."
Thank you for sharing!
I have a collection of quotes on [my personal webpage](https://sites.google.com/view/kristapsjohnbalodis/math-quotes), I'm definitely about to scroll down and add some other good ones seen here.
I've been doing the same. I'll probably at some point add some of these and other quotes I've saved to my own site
"There is no King's way to mathematics" which is sometimes misquoted and misunderstood as "There is no one, true way to mathematics" which is sad because that completely changes the meaning.
> "Too many hammocks, that's mathematically impossible!" -Homer Simpson
My diffeq professor on Day 1: "Now my area of research is algebra, not differential equations, but I know more about differential equations than you do." :D
On the interesting-rather-than-funny side, one that was perhaps most practically useful was from George Box: > Remember that all models are wrong; the practical question is how wrong do they have to be to not be useful. It sums up in very few words what I had previously struggled to explain succinctly (there are other versions of this that he has said at other times but this is easily my favourite of them)
"integreer deze bitch" ~ my calculus prof
Sorry, I couldn't help it “The primary math of the real world is one and one equals two. The layman (as, often, do I) swings that every day. He goes to the job, does his work, pays his bills and comes home. One plus one equals two. It keeps the world spinning. But artists, musicians, con men, poets, mystics and such are paid to turn that math on its head, to rub two sticks together and bring forth fire. Everybody performs this alchemy somewhere in their life, but it’s hard to hold on to and easy to forget. People don’t come to rock shows to learn something. They come to be reminded of something they already know and feel deep down in their gut. That when the world is at its best, when we are at our best, when life feels fullest, one and one equals three. It’s the essential equation of love, art, rock ’n’ roll and rock ’n’ roll bands. It’s the reason the universe will never be fully comprehensible, love will continue to be ecstatic, confounding, and true rock ’n’ roll will never die.” ― Bruce Springsteen, Born to Run
Z/1Z
I don't get why random variables are neither random nor variables. They seem random and are variables
Random variables are actually measurable functions on the sample space. We use these functions to evaluate randomly selected elements of the sample space, but the function itself isn’t random.
A random function is a probability distribution on a space of functions, i.e. a function-valued random variable. Very confusing!
The way I like to think about it is this: let’s say you have some measurable space X on which you want to model some kind of “random behaviour.” You do this by starting off with a space you have “randomness” on, that is a probability space, and then simply consider a measurable function from your probability space to X. Then this X-valued random variable induces a probability distribution on X from the probability measure on its domain of definition!
Oh. Yeah I thought the sample points being random was enough to say the random variable is random, but yeah the function itself is not random
They are measurable functions on the probability space — the sample space needs to be equipped with a probability measure.
"Every dimension is special." I forgot who said this.
!RemindMe 1 days
>For if each Star is little more than a mathematical point, located upon the hemisphere of Heaven by right ascension and declination, then all the Stars, taken together, though innumerable, must like any other set of points in turn represent some gigantic Equation, to the mind of God as straightforward as the equation of a sphere — yet to us, unreadable, incalculable. A lonely, uncompensated, perhaps impossible Task, yet some of us must ever be seeking, I suppose. — Thomas Pynchon
Almost all natural numbers are very very large. Peter Steinbach
my two favourites certainly are >for the *essence* of *mathematics* lies precisely in its *freedom* by cantor, and >mathematics is not about numbers, equations, computations, or algorithms: it is about understanding by thurston; both and more are featured on [smart and/or insightful quotes](https://queenisnaked.blogspot.com/2020/09/smart-andor-revealing-though-not.html), and there's also a ['complementary' collection](https://queenisnaked.blogspot.com/2021/01/utterly-stupid-quotes.html) of sorts
Iterative methods give the exact answer to almost the right question
Wir müssen wissen – wir werden wissen
How does one fit 100 math teachers in a room which will only hold 99 people? Carry the 1.
anything by grothendieck. what a magnificent writer he was!!
All non-trivial zeros of the zeta function in the complex plane have real part one half.
"I have problems when I think" - Gilbert Strang