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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
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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
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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,
479:
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,
401:
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
417:
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
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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
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the main definitions of mathematics found in the literature without making
Knowledge 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.
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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.
597:
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.
397:
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
489:
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
393:
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
642:
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
549:
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
593:
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.
602:
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"?
616:
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
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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).
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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.
667:. 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
574:
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? ... --
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on
Knowledge. If you would like to support the project, please visit the project page, where you can get more details on how you can help, and where you can join the
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that tell about legal and economic consequences of different definitions of mathematics? If so, that sounds salient enough to belong in this article. —
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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.)
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Well, if you can find the source, and it is reputable (say, by an established mathematician), then include it.
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Here a proper definition of "mathematics" is of great importance. An incorrect interpretation may really
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Information, for each major definition, on why it hasn't been accepted (that is, the main objections)
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Instead of having edit wars and arguments about which definition belongs in the first sentence of
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on
Knowledge. If you would like to participate, please visit the project page, where you can join
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I've always heard that most mathematicians were platonists, though I doubt this is verifiable.
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I don't have any sources handy at the moment, but I have the following responses/points:
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The 1933 OED definition is actually
Kantian. Let's put up Kant's definition explicitly.
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Brief explanations of how each definition relates to the broader school of thought
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what
Knowledge is. Please just do a wonderful job summarizing the literature.
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The associated definitions of fundamental mathematical concepts like "number"
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Still think a name change might help clarify the articles intent - perhaps
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Thanks. It wasn't obvious from the article what direction this was heading.
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I'll try to add some stuff about Platonism/Mathematical Realism on monday
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saying logic is math, etc. so it would be good to discuss the matter.)
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Naturally, please do not post your own opinions about math. That is
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More definitions of mathematics from more schools of thought
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which becomes a circular definition. Clearly we recognize
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If stays here, should it be classified/renamed as a list (
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Many thanks for your efforts to make this a great page.
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