Talk:Definitions of mathematics

Context
Instead of having edit wars and arguments about which definition belongs in the first sentence of mathematics, or which is the true definition of mathematics, please improve this page by explaining all the main definitions of mathematics found in the literature without making Wikipedia take a side in the controversy. Let's convert our differences of opinion into a really good introduction to the opposing definitions of mathematics that any lay reader can understand and find valuable.

To do
Here are some things that this article could use:


 * More definitions of mathematics from more schools of thought
 * Information, for each major definition, on why it hasn't been accepted (that is, the main objections)
 * Brief explanations of how each definition relates to the broader school of thought
 * The associated definitions of fundamental mathematical concepts like "number"
 * The 1933 OED definition is actually Kantian. Let's put up Kant's definition explicitly.
 * Sourcing on just how controversial the definitions of mathematics really are. For example, has anyone conducted polls of mathematicians? I have heard repeatedly that most practicing mathematicians don't care about the definition of mathematics. If anyone has a source on that, please post it.
 * A nicer-looking format

Naturally, please do not post your own opinions about math. That is not what Wikipedia is. Please just do a wonderful job summarizing the literature.

Many thanks for your efforts to make this a great page.

--Ben Kovitz (talk) 15:23, 22 February 2009 (UTC)

I don't have any sources handy at the moment, but I have the following responses/points:
 * I've always heard that most mathematicians were platonists, though I doubt this is verifiable.
 * I'll try to add some stuff about Platonism/Mathematical Realism on monday
 * I recall reading, but can't remember where, an article about how the differences between representations of numbers as sets presents a problem for realists, I don't think it does, but if anyone knows what I'm talking about, it would be an interesting criticsim.
 * As for format, perhaps we could discuss each perspective under a different heading presenting a run down of what the view is, its history, arguments for it, and ending with some criticisms. This would organize everything coherently and give each concept equal prominenance.
 * Finally, by "mathematics" do we mean using any formal system, or are we talking modern mathematics; relatedly, it would be good if we could find some material that helps draw a line between exactly what is logic and exactly what is mathematics(I've read articles saying math was just logic, others saying logic is math, etc. so it would be good to discuss the matter.) Phoenix1177 (talk) 17:46, 28 February 2009 (UTC)

I look forward to seeing the information that you find, Phoenix. A known problem involving representations of numbers under one definition that doesn't occur under another definition would be fantastic. That would give a reader a really clear idea of what the controversy is about. I think "any formal system" is itself one of the definitions of mathematics (already there as "formalism", though not in those words). We should include the most common definitions, modern definitions, ancient definitions, and even some unpopular definitions at the end, all ideally in the words of the people who originated them. Hopefully before long, it will make sense to link to this page from the much-fought-over first paragraph of Mathematics, so we can cover the many opposing definitions without taking a side and with no original research. —Ben Kovitz (talk) 10:10, 1 March 2009 (UTC)

Wikipedia vs Wiktionary
Seems like this content might make more sense in Wiktionary. The entry there doesn't have near the depth that this does.

If stays here, should it be classified/renamed as a list (List of definitions of mathematics). If extends beyond a list, how differentiate from something like Philosophy of mathematics? Zodon (talk) 20:59, 1 March 2009 (UTC)


 * Let's give this a few weeks to develop. So far, only one person (me) has edited the body of the article. I envision it as a coherent article, including history and comparisons of ideas, not just a list (see "To do", above). Of course, it might develop into something different than what I envision, and that's OK. It's different from Philosophy of mathematics simply because it focuses on definitions, a common topic of argument, often by people who are unaware of how old the controversy is and the main lines of thought that have developed.


 * Here's why I don't think it belongs on Wiktionary: this article isn't trying to define different senses of the word "mathematics", it's comparing and contrasting the many proposals and controversies over how best to define mathematics (the thing, not the word).


 * —Ben Kovitz (talk) 14:54, 2 March 2009 (UTC)


 * Thanks. It wasn't obvious from the article what direction this was heading.
 * Still think a name change might help clarify the articles intent - perhaps Comparison of definitions of mathematics, or just Definition of mathematics?  (it was the Definitions that suggested this was a list).  Or what is the matter called in the field?
 * Seems like some of the current content might be readily copied/adapted to wiktionary to augment the entry there. (Doesn't mean have to remove here.) Zodon (talk) 23:39, 2 March 2009 (UTC)

What is mathematics?
I saw somewhere on the web, which I can't find now, a quotation which said: "Mathematics is the study of any and all absolute truth." ... but then there's Probability ... approximation ... multi-valued logic ... :-p Charvest (talk) 17:40, 3 March 2009 (UTC)


 * Well, if you can find the source, and it is reputable (say, by an established mathematician), then include it.

What I find remarkable is that there is no question what the domains of math are (perhaps some fuzziness over pure versus applied, but that's all). And yet, though mathematicians agree on what is and what isn't math, no one seems to be able to give a clear definition, other than listing the domains, which becomes a circular definition. Clearly we recognize something that clearly and unarguably unifies all of math. We all recognize it when we see it. "This is math. This is not." Why can't we put our finger on it? Or has someone? ... --seberle (talk) 02:04, 30 October 2009 (UTC)

The definition of definition
Perhaps it seems an endless recursion issue at first sight, but I do believe it is helpful to consider the definition of definition for a moment. Or, in more practical terms, to distinguish types of definitions.

Ontological definitions try to grasp the essence of something. That is always hard, and for maths it may be harder than for anything else. But there are also teleogical definitions, used e.g. by lawyers for a specific purpose. And there may be a colloquial definiton.

The European Patent Convention (EPC) excludes mathematical methods (only "as such", but that is a different debate). The rules of the law of treaties (as laid down in the Vienna Convention on that topic) phraes the following obligation: "A treaty shall be interpreted in good faith in accordance with the ordinary meaning to be given to the terms of the treaty in their context and in the light of its object and purpose." (art. 31(1)). So what is the "ordinary meaning" of "mathematical methods"?

Here a proper definition of "mathematics" is of great importance. An incorrect interpretation may really ruin real people. While lawyers often solve definition problems by adopting their own definition, in this case the law of treaties does not allow that (the EPC is an international treaty).

US courts for some time distinguished mathematical from non-mathematical algorithms. For mathematicians this distinction may be preposterous, but a lower court actually was compelled to make that distinction in order to be able to reconcile the Supreme Court pecedent with basic logic.

In 1998, the (alleged) "mathematical algorithm exception" in US patent law was abolished (or actually denied). The result is a tsunami of crazy patents (see Ben Klemens, Math You Can't Use). So some rule for mathematics seems appropriate ... Rbakels (talk) 09:50, 13 August 2010 (UTC)


 * Rbakels, these are all fine points, but it's not Wikipedia's place to make a ruling on matters of controversy. Hopefully we can do a good job telling the main definitions of mathematics that have been advanced, and the main reasons people have given for them. On the other hand, are you aware of any  sources that tell about legal and economic consequences of different definitions of mathematics? If so, that sounds salient enough to belong in this article. —Ben Kovitz (talk) 15:58, 11 March 2011 (UTC)


 * I agree with Ben Kovitz. I think a discussion of the legal status and economic consequences of the definition of mathematics could be an important addition to this article, as long as it is kept NPV. Please feel free to add such a section, Rbakels. --seberle (talk) 16:33, 11 March 2011 (UTC)