What makes generalization of a mathematical notion correct?

Question

As far as I know, the notion of open set has been formulated after the notion of open interval in the real line. Was the goal of generalization to allow the definition to work in higher dimensions?

A subset U of the Euclidean n-space R^n is called open if, given any point x in U, there exists a real number ε > 0 such that, given any point y in R^n whose Euclidean distance from x is smaller than ε, y also belongs to U.

When generalizing from any notion A (in this example, open interval) to a notion B (here - open set), our goal is to ensure that if A is true, then B is true as well. So we want to define a generalization of open interval in such a way that if an interval is open, then it's an open set. Obviously, we don't require the converse to be true.

Is it the only definition that qualifies as generalization of "open interval"? I could find some other definition of open set that would satisfy the condition that if something is an open interval, then it's an open set (remeber, the definiton of open set would be different here). What made mathematicians agree on this particular generalization?

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To give another example, consider the definition of continuous function given by Cauchy. Function f is continuous at x, if for all ε > 0, there exists some δ > 0, such that if y is no further than δ from x, then f(y) is no further than ε from f(x).

A more general definition used in topology is (X and Y are topological spaces):

A function f :X→Y is continuous if for each open subset V of Y, the set f^{-1}(V) is an open subset of X. f^{-1}(V) means the set of all points x of X for which f(x) ∈ V.

This sounds very general and very abstract. It makes me ask - how abstract a definition (axiom) can be, and still be "valid"? As far as I know, definitions are axioms in modern mathematics. Axioms are supposed to be obvious, taken as true without proof. Yet, some definitions (axioms) are better motivated than others. To remind my earlier point - I could just as well find a different generalization of Cauchy's continuous function, such that if a function was continuous according to Cauchy, then it would be continuous according to the new, different definition.

It follows that mathematicians are still formulating new axioms, by creating generalizations of definitions such as those mentioned above. They can informally justify them, but not prove them (well, that's why they are called axioms). But somehow, some axioms (definitions) are more true than others. We're trying to measure the correctness of a definition by the strength of its motivation, justification, which are informal considerations anyway.

In summary:

How are generalizations created, justified and accepted? Could we have chosen different generalizations (I'm suggesting that we could). Is every generalization a creation of another axiom? Is there the only, ultimate, correct and valid generalization?

Answer 1

How are generalizations created, justified and accepted?

Often, it's what benefits mathematical practice. For instance, unifying fields of study and discovering interesting results are considered beneficial, and this desire can drive a lot of the attempt to find generalizations, and can be used as a criteria to justify and accept these generalizations.

Could we have chosen different generalizations (I'm suggesting that we could).

Yes.

Is every generalization a creation of another axiom?

No. Some generalizations simply discover new results, others are isomorphisms among pre-existing fields, still others could be different formalisms.

Is there the only, ultimate, correct and valid generalization?

If you're a Platonist, perhaps, but personally I think not. Even finding the one generalization that unifies all others may not qualify as some generalizations facilitate certain practices at the expense of others. Since different mathematics have different goals, it stands to reason that different generalizations will be adopted as the needs dictate.

Answer 2

A definition in mathematics is generalized only if doing so proves useful and enlightening in some way.

Regarding your example of an open set, it is not necessarily the idea of an open set that is generalized, but the notion of a topological space. The concept of a topological space proves to be fundamental in many contexts so it is desirable to have a deep understanding of its properties.

Often notions in mathematics are generalized from those in the context of the real number line and Euclidean space in general - for instance the notion of a topological space - in a way that the definition reduces to the familiar object on the real line or in Euclidean space. But this need not always be the case.

There is no 'level of abstraction' beyond which a generalization would cease to be 'valid' or 'correct'. The only criteria is usefulness.

In general a definition is only a notion in mathematics that is deemed to be so useful and important that it is worth distinguishing by giving it a name. Usually the importance is due to the notion and use of its properties turning up in many contexts.

For instance in many contexts we need to use or draw upon the properties of a topological space in order to prove other results. The extent to which this scenario occurs over and over again determines the usefulness of a definition and/or it's generalization.

An axiom is something different to a definition. An axiom is something taken or assumed to hold or be true. A definition is just the formal assignment of a name to a mathematical concept.

Answer 3

Good question; it relates to the notion of judgement, which for Kant is part of aesthetics; which shows that aesthetics, when philosophically thought, isn't the conventional notion - that of art; it is of course related in some sense.

Though you've asked about how the right definition is chosen in mathematics, similar questions can asked elsewhere; for example, why does a novelist choose one word over another? or an architect one material over another? the choice of artists and artisanry is deliberate; because where craftsmanship is exercised, judgements must be made; and these judgements become encoded in the tradition of that art or craft; and in this sense, overtly objective disciplines, such as physics and mathematics, as well as being a science, are also arts; that is crafts.

This is Grothendiecks famous description of his approach to mathematics:

The unknown thing to be known appeared to me as a stretch of earth, or hard marl, resisting penetration...the sea advances insensibly, in silence; nothing seems to happen, nothing moves; the water is so far off you hardly hear anything; yet it finally penetrates the resistant substance.

And of course, mathematicians talk about good mathematics, or beautiful mathematics or what ought to be the case (the Kantian triple, in a sense; but mostly not) and this, to some extent, justifies the description above.

It is one of the purposes of a good mathematical education to enstill these values - usually called mathematical maturity - which are in a sense, unencodable, into the human substance; hence the institutions and traditions of this craft.

To come back to your specific question; higher-dimensional geometry was developed post the advent of coordinates by Descartes, (that 'honest' thinker in Kierkegaards phrase, who used doubt as a tool and not as a scalpel - critique rather than criticism); in that language it's easy to generalise that most basic of identities - the one of Pythagoras; no visualisation is necessary as would have been in the geometric imagination; one can then push through calculus in this context using the technology of vector spaces and then manifolds; in both cases, there is a natural metric, a natural way of measuring distance, which is related at a basic level to the Pythagorean theorem; in a different direction, metric spaces began to be investigated; these are just pure spaces with a different but related notion of distance, one even more basic than Pythagoras - I mean the triangle identity; it's hard to argue with, since it seems so natural; and so basic; and thus one might argue this ought to be enough; but it was noticed in many arguments that the metric wasn't necessary and one could use open sets; this is an important innovation, because it drops the measuring rod and instead, for lack of a better word, uses 'place', the word used by Aristotle in his investigation of space; in a way it measures space by space itself - the measurement is commensurable (like changes like, so like measures like).

Aristotle, in his physics, linked this notion, Place, to ideas of change and variation, contiguity and continuity, and of the parts to the whole - all important notions.

It's important to link the development of these notions to others; topology as a discipline wouldn't have been conceivable; or rather, it could be conceived as a Concept, and it was done so; but it couldn't be determined as an Idea (to steal a term from Hegel); that is placed in sharp relief in the axiomatic manner without a certain amount of technology; and that technology is what is now called Set Theory; and this Notion carries the basic idea that things are made of elements, of points. This is the modern view; the post-modern view is that extensionless points aren't neccessary, or are wrong, which was the view argued by Aristotle, in his investigation of continua; one can in fact, drop the points of a topological space, and just investigate its cohesive structure - it's topology - thus locale theory, or in a phrase that should be more famous than it is, 'pointless topology'.

One can go further, homotopy theory is a sub-discipline in topology; and this uses the notion of paths; after much development, and many results; it was noticed that the subject could be 'axiomatised', and this is called abstract homotopy theory, and one of its main tools is 'model categories'; then in an another innovation, it was noticed that this was a 'coordinatisation' of infinity-categories: for not only are there paths between points, there are paths between paths; this again is an ancient observation, that can be traced to Parmenides, 'doesn't each notion of likeness, give rise to another, and then again'.

Still, this shows that open sets are only one step, though a good strong step, in the long tradition of investigating space and continuity.

Finally it's worth pointing out, that historically speaking that open sets were not the only way, then of tackling continuity; given that limits by sequences are the basic tool in analysis, one might expect it would be this tool that would be generalised; and this is possible - they are called nets, and not widely used; they have a dual form - filters - and these are used more widely; but open sets introduces a new idea, or rather return to us an old one, that space is cohesive, and that it can be 'unglued' into small pieces - note, that open sets have extension, that they are not points, and that their extension is not measured by a number - and then 'glued' back together; and it is this notion of 'gluing' which generalises to a more general context like sheaves and stacks.

One might do without the 'Judgement of God' (and it sometimes appears that in this world one must); but one cannot do without Judgement altogether: culture, in its widest sense, being essential to man, in part his essence objectified, is also the objective manifestation of judgements; also too, in its widest sense.

In Little Gidding, a poem by TS Eliot, he writes:

We shall not cease from exploration

And the end of all our exploring

Will be to arrive where we started

And know the place for the first time

and this is the repeated motion of the mathematical mind meditating on space; broadening and deepening it; by repetition and difference; but each time returning to the beginning and seeing the same ground, again; the same earth, again; as though for the first time, afresh; and thus -

To become renewed, transfigured in another pattern