17
$\begingroup$

I have to write a short monograph as an assignment for a course on the philosophy of science. Being a math student, of course I want to opt for something math-related. After some initial ideas which would have needed way too much research, I imagined I could narrow it down to a question which I have always wondered about: is mathematics discovered or created?

I'm thus asking for references to books/papers/quotes/anything which adresses this question. I hope it is not too soft for a math.SE question; I apologize if it is.

In particular, I remember a quote saying something like "Natural numbers were created by God. All else is the work of men", I'd like to know its exact statement and author.

Anything, even if tangentially related, may come in handy. Thank you.

  • 7
    "God made the natural numbers; all else is the work of man." -- Leopold Kronecker2011-06-25
  • 0
    I see different quotes in different places though (I come across both natural numbers and integers, hmm)2011-06-25
  • 8
    As a Math student, I think it is best if you don't do a project on Math.2011-06-25
  • 0
    @Bruce Stonek: Interesting topic. Maybe too close to things that philosophers actually know something about.2011-06-25
  • 0
    @user6312: thank you. Also, lol @ "Bruce"!2011-06-25
  • 4
    I agree with jspecter. To be sure, it's totally up to you, but by doing mathematics anyway one gets some exposure to topics in the philosophy of mathematics. On the other hand, most actual sciences are quite different from mathematics in the way they operate (as a zeroth order approximation one might say mathematics is deductive whereas real sciences are inductive) so taking a course on philosophy of science seems like a good opportunity to learn about non-mathematical science.2011-06-25
  • 4
    I also have to say this: if the assignment is to write a research paper, presumably a big part of this assignment is to look through the literature yourself. Asking for help from (an audience which includes some) much more experienced people at the very beginning doesn't seem in the spirit of the assignment: why don't you start looking yourself?2011-06-25
  • 0
    @Pete: it is not a research paper what I have to write (if I understand well what you mean by "research paper"), just a 10-15 pages survey on some topic of interest. To be honest, the course is not on philosophy of science but rather on "university, science and society". Some of the early lectures were on the history of science and some epistemology; based on that I want to write something related to that part of the course, and not the more socially-oriented part which is not of great interest to me. In fact, I've matriculated to this course only because it is compulsory and there is no...2011-06-25
  • 2
    ...proper "philosophy of math/science" or "epistemology" course available. So if I have to do some work for a course that doesn't interest me much and is just compulsory, I might as well do it in something that interests me, is what I think.2011-06-25
  • 7
    This exact question is addressed at http://philosophy.stackexchange.com/questions/1/was-mathematics-invented-or-discovered2011-06-25
  • 1
    The question you seek references on (whether mathematics is discovered or created) is neither coherent nor meaningful, and should be taken to the philosophy section of stackexchange. It is also essentially answered by http://math.stackexchange.com/questions/30572/good-books-on-philosophy-of-mathematics2011-06-25
  • 2
    To say a mathematical object exists is to say it is logically possible for affairs to exemplify its structure. Logical possibility exists, by assumption, independent of the human mind, and hence mathematical objects are discovered. On the other hand, we may say an idea is invented if it is creatively fashioned from other, previously known concepts in a meaningful way. Therefore mathematical objects are also created. So long as discovery and invention hold these straightforward definitions, we must conclude mathematics is a process of both simultaneously. Those are my thoughts on the question.2011-06-26
  • 5
    @Bruno: No, I mean research in the sense of "library research" not "mathematical research" or "original research". The fact that the course is required and that you don't seem to be thrilled about taking it is not really an adequate excuse for asking the internet mathematical community to help you write it. I am voting to close.2011-06-26
  • 3
    @Pete: But Pete, I'm not asking for help in *writing* the text, not at all. I'm just asking for useful references. What I'm supposed to compose will not be comprised merely of a bibliography. I don't see how this is unethical, if that's what you're implying.2011-06-26
  • 4
    @Bruno: Pete is saying that library research is an important aspect of writing a research paper - and part of the educational value of a research paper assignment is learning how to do library research *on your own*. We would be depriving you of the experience of doing library research by providing you with a list of resources.2011-06-26
  • 6
    I really don't understand the votes to close this question as off-topic - this is definitely a question relating to the history and development of mathematics. I therefore vote *against* closing following [this suggestion here](http://meta.math.stackexchange.com/questions/1869/exercising-the-vote-not-to-close). The next user who wants to cast a vote to close should leave a comment cancelling my vote instead of voting. (please vote this comment up so that it appears above the "fold")2011-06-26
  • 2
    @Zev: What's wrong with providing Bruno (Bruce) with places where he can start looking? Knowing Bruno from several interactions here and on MO I'd be really surprised if he chose the easy way and decide not to look beyond the suggestions provided here.2011-06-26
  • 0
    @Theo: thank you for your confidence.2011-06-26
  • 0
    @Theo, Bruno: I certainly wouldn't say it's unethical, and indeed I think there's an argument that "asking online" is now a proper part of modern library research; I would not vote to close this question (even if my vote were not binding). However, I am somewhat sympathetic to Pete's view, and Bruno's latest response to Pete did not seem to fully understand it, so I was helping to explain.2011-06-26
  • 1
    @Zev: Thank you for the clarification. I simply don't think Bruno asked out of laziness but looked for input and was honest about his motivation (or slight lack thereof), so I stand by my previous comments. I think it is a very good idea and an important discussion to be aware of and the platonism versus formalism versus intuitionism debate and its consequences had an impact on 20th century's mathematics that can't be neglected (I'm a bit astounded that these keywords were not mentioned so far - or only implicitly in a link).2011-06-26
  • 1
    @Bruno: I don't know if you saw it, but there is a deleted answer pointing you to Stanford encyclopedia entry on [Philosophy of Mathematics](http://plato.stanford.edu/entries/philosophy-mathematics/) and [this MSE thread](http://math.stackexchange.com/questions/30572/good-books-on-philosophy-of-mathematics). I haven't looked how helpful they actually are, but I thought it might be worth pointing it out to you.2011-06-26
  • 0
    @Theo: Thank you! No, I can't see deleted answers. I will surely check it out. The encyclopedia entry looks very interesting: I don't know if it deals with the subject at hand, but in any case it seems it will be a very illuminating reading.2011-06-26
  • 0
    FWIW, I ended up writing the essay on something else (also math-related, though); so for those who were reluctant to throw out some ideas, be at ease: the answers and comments "only" contributed to my knowledge of these profoundly interesting matters.2011-07-20

8 Answers 8

14

Original answer by trutheality:

Die ganzen Zahlen hat der liebe Gott gemacht, alles andere ist Menschenwerk.

-Leopold Kronecker

Translated to English:

God made the integers; all else is the work of man.

It also often appears as "natural numbers".

A quick search online suggests that "ganzen Zahlen" means integers in German. But I don't speak German, so any input from someone who does is appreciated.


Added: (Theo Buehler)

Kronecker's quote is from a talk he gave at the "Berliner Naturforscher-Versammlung" in 1886. I'm not aware of a transcript of this talk. The quote is most often cited in the form in which it appears in the very interesting obituary by H. Weber:

Excerpt from Weber's obituary

The obituary can be found in the Jahresbericht der Deutschen Mathematiker-Vereinigung Vol. 2, (1891/92), the quote is on page 19.

Here's an attempt at a translation (rather loose):

Concerning the rigor of notions [Kronecker] imposes highest requirements and tries to squeeze everything that should have a right of citizenship in Mathematics into the crystal clear and edgy form of number theory. Many among you will remember the dictum he made during a talk at the 1886 reunion of natural scientists in Berlin ("Berliner Naturforscher-Versammlung"): "God made the integers; all else is the work of man."

  • 8
    I don't understand the down-vote. It answered what the OP asked, and it's a great answer in its own right, so I'm up-voting this answer.2011-06-25
  • 1
    By "Die ganzen Zahlen" one indeed means $\mathbb{Z}$, while $\mathbb{N}$ is most often called "die Menge der _natürlichen_ Zahlen". The word "ganz" is synonym for "komplett", "unbeschädigt" which you could translate as "integer". (You never say "die unbeschädigten Zahlen", though.)2011-06-25
  • 0
    @leftaroundabout Thanks. Feel free to add that by editing the answer if you'd like.2011-06-26
  • 0
    @leftaroundabout: What you're saying is true in nowadays usage, but I'm not sure that this applies to the present discussion (lacking context).2011-06-26
  • 0
    Thank you all for your research on such an informative answer!2011-06-26
  • 1
    @Bruno: I added a loose translation of the passage I displayed. I attempted to retain the colorful language, I hope you can follow the idea.2011-06-26
  • 0
    @Mike: I'm upvoting the answer AND your comment :)2011-09-30
11

I would like to recommend 'The Two Cultures of Mathematics' by W. T. Gowers http://www.dpmms.cam.ac.uk/~wtg10/2cultures.pdf

In the setting of this article, personally, I prefer to say, Theory is created, while a solution to a math problem is discovered.

  • 0
    I fixed your link. I know of this article and have read it before (and liked it a lot). It is of course related. Thank you.2011-06-25
6

As a physicist who has recently switched to a Mathematics career, I can give you only my opinion based on my experience and knowledge of the Laws of Nature. I do believe mathematics is completely real and is discovered not invented. A similar opinion was held by physicist Richard Feynman, in particular I recommend you watch his old lectures on the Character of the Physical Law, concretely lecture no. 2 about "The Relation of Mathematics and Physics" to appreciate that mathematics seems to be the proper setting to talk about the structures we find in Nature.

If you want to deepen about the mathematical universe hypothesis concerning the (for many crazy) idea that everything is mathematical, see the preprint by Max Tegmark and his other articles in his website.


(This answer contained an excessively long digression about those ideas but I have removed it in order not to contribute to endless debates; only the previous references remain as useful).


  • 1
    I can't claim to have read your entire post in detail, but I would like to respond to the claim "Anything we will ever be able to say about Nature will be mathematical in the end." This obviously cannot be known - while it has been our *experience so far* that natural phenomena can be described mathematically, there is no *reason* why the universe should be able to be described by math, or indeed by anything. As Einstein said, "The most incomprehensible thing about the world is that it is at all comprehensible", and we can never rule out that the universe is **not** completely comprehensible.2011-06-26
  • 1
    Whereas there are lots of reasons to *"believe"* (scientifically by induction) that Nature is indeed mathematical (name every single structure and mathematical law I mentioned and all the rest), there is NO single reason to believe the contrary except the logical possibility of it. All the evidence supports the claim so far. We cannot be sure, of course, but THAT IS SCIENCE, as we cannot be sure that a relativistic black hole passes through the solar system and there is no rising sun tomorrow, all the evidence says it is quite improbable...2011-06-26
  • 0
    Besides, there is a school of mathematical thought which says that natural languages can be reduced to mathematics in the end. Any information-theoretic construction that we made, be it in mathematics or in any other language, is in the end some kind of structure within the correlations of the information in our brains and the degrees of freedom perceived from the outside world. Since everything that can be said about Nature must be said in some language, everything that an inside observer will ever say will be an information-structure, and thus reducible to formal systems and mathematics.2011-06-26
  • 0
    @Zev Chonoles: as I said in the post, it is my opinion based on my experience and knowledge and I hoped not to open a debate. To rebut some of your claims: most mathematicians blind themselves to the scientific method because they want complete truth, but the example of physics gives you the answer that INDEED Nature is described by Math to the level of precision we have today. The amount of progress made by that knowledge PROVES it is described at some level by math.2011-06-26
  • 0
    @Zev Chonoles (cont'): *We can never rule out anything?* are you sure? I though science made advances because of falsifiable models of Nature. As Feynman says in the lectures I linked to, "we can only be sure about what is false". I challenge anybody to defy gravity and jump a cliff... since you could not be sure if it starts repelling that precise moment. Some things are true and those are here to stay. One of them is the usefulness of mathematics in the natural sciences... of that you can be sure.2011-06-26
  • 0
    @Zev (end'): "*we can never rule out that the universe is not completely comprehensible?*" If it is not completely comprehensible for any intelligent being inside Nature, then whatever that it is comprehensible will be some kind of informational correlation between knowledge and patters perceived from Nature. As Wittgenstein said "of what you cannot talk it is better to be silent", therefore whatever is not comprehensible cannot be talked about in any language. Einstein's quote is of a more sentimental nature than epistemological, expressing the amazing fact that it indeed seems comprehensible2011-06-26
  • 0
    @Javier: The statement that any "natural language" or "information-theoretic construction" "can be reduced to mathematics in the end" is completely contentless - barring some sort of actual definition of these terms (which seems unlikely to me), one can simply change one's meaning in such a way that anything that can't be reduced to mathematics isn't a "natural language", etc.2011-06-26
  • 0
    Also, I only contested your inductive claim about the usefulness of mathematics in the natural sciences, not the actual inductive reasoning necessary to do physics. There is an extreme difference between "We've done an experiment 100 times, it came out the same every time, *we will conclude that it will happen that way every time*" and "We've examined 100 physical phenomena, they were all able to be described mathematically, *we will conclude that all physical phenomena can be described mathematically*."2011-06-26
  • 0
    @JavierÁlvarez let us [continue this discussion in chat](http://chat.stackexchange.com/rooms/633/discussion-between-zev-chonoles-and-javier-alvarez)2011-06-26
  • 0
    Javier Alvarez above made the typo of "though" for "thought" ("I though science made advances"...). This is an example of what I call a "dog-to-cat" error, which I discuss here at MSE, exactly here: http://meta.math.stackexchange.com/questions/2575/difference-between-mse-and-mo, and again I ask non-native English speakers of this site to tell us how much of problem such errors on the part of native speakers are for them.2011-09-30
  • 0
    @MikeJones: in my case, most of the time, it is just a keyboard mistyping "I thought science.." It seems to me that those kinds of errors may be confusing sometimes but I do not think it is a major issue for non-native speakers who are fluent enough (at least reading English math books). For example, and JUST in my personal opinion, I would consider more problematic sometimes the extent to which many discussions are done in the category theory framework (especially for people like me who comes from theoretical physics and regard algebraic geometry as geometry and not as abstract archery).2011-10-08
1

In his autobiography Un mathématicien aux prises avec le siècle L. Schwartz discusses the question and says that it somewhat complicated. I haven’t the book, so can't cite properly, but the reasoning was something like this. Consider, for example, complex numbers. They can be regarded as human invention. But all their properties then are discoveries.

1

An excellent discussion of these issues is given by Reuben Hersch in his book What is mathematics, really?. The general message is that mathematics is philosophically "humanist" - it has a socially created reality. This doesn't give much of an idea of what the book is about, but it's about the best account of these sorts of issues that I've seen.

1

Doug Hofstadter's book Fluid Concepts and Creative Analogies responds to this question. He adopts the metaphor of mathematician as a person feeling around in a dark cave. He feels that mathematicians use their creativity to discover natural truths.

(So, I guess his answer might be "Both"?)

1

Thomas Kuhn's book "The History of Scientific Revolutions" is basically a treatise on precisely this question. He specifically takes up this conundrum of discovery versus invention in terms of the discovery/invention of oxygen.

0

The operative word is research. It is a search of the truth about things that already exist.

About theories - they require proofs that must be acceptable by fellow mathematicians. The sufficiency of a proof is subjective and varies with the person and time. This itself makes theories more of a hypothesis.

I came across a quote (by someone - I do not remember) - We call the theories we believe axioms and the facts we disbelieve theories.