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"Everyone knows what a curve is, until he has studied enough mathematics to become confused through the countless number of possible exceptions."

Felix Klein

What notions are used but not clearly defined in modern mathematics?


To clarify further what is the purpose of the question following is another quote by M. Emerton:

"It is worth drawing out the idea that even in contemporary mathematics there are notions which (so far) escape rigorous definition, but which nevertheless have substantial mathematical content, and allow people to make computations and draw conclusions that are otherwise out of reach."

The question is about examples for such notions.

The question was asked by Kakaz

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    $\begingroup$ In mathematics, by mathematicians. Everything is clear? I suppose mathematics is still live nowadays... $\endgroup$
    – kakaz
    Feb 25, 2011 at 21:08
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    $\begingroup$ "Everything is well defined in modern mathematics" - We really don´t know that for sure yet (i.e. consistency of ZFC)... "Mathematics is more about correctness than about truth." -I would argue that it is more about the relative truth. Being about correctness is contains too much of a self-purpose... $\endgroup$
    – M.G.
    Feb 26, 2011 at 1:55
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    $\begingroup$ Qiaochu - I suppose than natural language meaning is enough for "notion" and "used". If You are in trouble You may refer to nouns present more that 10 times in books from LCC classification, class Q, subclass QA, from last 60 years, as a names referring mathematical objects ( that is other than common names for things, people, and animals or plants. By "thing" I mean any part of physical reality which may be observed). $\endgroup$
    – kakaz
    Feb 27, 2011 at 10:30
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    $\begingroup$ @Qiaochu - I understand Your position, but I think nobody have to fear vagueness in such situation. It is just another soft-question on math-overflow. It is some fun. I know that You are professionals but I am an amateur. I would like to play with mathematics. Usually it is worth of mention what do we use without proper definition, fighting between intuition and complicated formalism, and possibly why. I do not understand why question which is obviously interesting and have potential to broaden horizons for many people is so controversial. $\endgroup$
    – kakaz
    Feb 27, 2011 at 19:05
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    $\begingroup$ Closing this thread seems more like punishing someone for being an amateur rather than enhancing the quality of the site. Now and throughout history, I believe, a large percentage of the most interesting mathematics revolves precisely around those notions that are used but not (yet) clearly defined. A big list of such subjects seems extremely valuable to me. Vote to reopen. $\endgroup$ Mar 1, 2011 at 16:01

32 Answers 32

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Notion of calculability:

A function of positive integers is calculable only if recursive.

Calculable function ( in a objective meaning) as used in Church-Turing Thesis http://plato.stanford.edu/entries/church-turing/

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    $\begingroup$ My view of this is the opposite: the beauty of the Church-Turing thesis is that it does give a precisely defined and widely agreed upon meaning to "effectively calculable". From a mathematical perspective, the thesis is itself a definition, which takes a somewhat vague notion (anything that can be computed by any systematic method or algorithm) and equates to it a rigorous mathematical concept (recursive functions). $\endgroup$
    – Henry Cohn
    Feb 26, 2011 at 15:01
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    $\begingroup$ I do think the busy beaver function isn't effectively calculable in the intuitive sense, as well as the technical sense. It's a well-defined function, but we have no algorithm for actually computing it. We do have an algorithm for proving lower bounds that will eventually converge to the true answer for any given case, but the convergence is incredibly slow (there is no computable upper bound on the time to convergence) and there is no way of knowing when convergence has happened. Being able to recognize when you have arrived at the answer seems like an essential property of algorithms. $\endgroup$
    – Henry Cohn
    Feb 26, 2011 at 17:10
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    $\begingroup$ kakaz: There is no algorithm for computing the busy beaver function, you are misunderstanding a side remark in that wikipedia link. You are also misunderstanding the Church-Turing thesis. When we use notions such as "effectively computable" we mean the formal notions, not some vague intuitions. As Henry is pointing out, the Church thesis, which is not mathematics, can be seen as a definition. $\endgroup$ Feb 26, 2011 at 17:34
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    $\begingroup$ Note that the Church-Turing thesis could almost be regarded as a statement of physics - the laws of physics of this universe only permit computation of recursive functions, but it is conceivable that other possible universes could do better. I.e. it would be the statement recursive <=> physics-computable. I say "almost", however, because I don't think anyone has actually come up with a precise notion of physics-computable! $\endgroup$ Feb 27, 2011 at 0:43
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I think that the widely-used concept of concepts being clearly defined is not clearly defined.

For example: How could one decide whether a single "concept" is "clearly defined" or not? If one could, I would argue that then all of modern mathematics would be not clearly defined. Already the concept of a set seems to lack a precise definition, and even lack the possibility to be defined precisely. Being clearly defined therefore seems to me at best like a vague comparative notion. For example, we could say that we regard some concept as clearly defined if its definition is as clear as the definition of a set, whatever "as clear" means in this context...

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    $\begingroup$ This seems to me to be a poor answer, for the same reasons I gave in a comment on the answer by Buschi Sergio. $\endgroup$
    – user21349
    Jun 21, 2015 at 18:50
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