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I have some trouble doing standard computations in calculus because of the notion of a differential, otherwise known as an infinitesimal, being rather ill defined, in my experience.

Are there any fields of mathematics that someone can recommend for trying to come to a more rigorous grip on the notion of infinitesimals? I've heard and read up about non-standard analysis, but from what I can tell, even the rigour of non-standard analysis isn't as firm as that of more established branches of mathematics. How accurate is this perception?

Any help is appreciated, thank you.

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    $\begingroup$ "even the rigour of non-standard analysis isn't as firm as that of more established branches of mathematics" what makes you say that? Non-standard analysis is as rigorous as it gets. $\endgroup$ – wilkersmon Feb 17 '18 at 9:41
  • $\begingroup$ It's only from what I can tell. There are sections of webpages devoted to Non-standard analysis that revolve around criticisms of it, something that I don't often see surrounding other branches of mathematics. But then again, that is why I am here, to learn. Thanks for the input @wilkersmon $\endgroup$ – joshuaheckroodt Feb 17 '18 at 9:47
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    $\begingroup$ This is really just a comment since the body of your question makes it clear you just care about non-standard analysis, but there are lots of other contexts with infinitesimals too, which makes the question in the title have a very different answer. $\endgroup$ – Mark S. Feb 17 '18 at 13:46
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    $\begingroup$ Also if you have some moral/philosophical issues about infinitesimals, or even infinite sized objects, you need to get over them. Mathematics isn't about if it's possible in a laboratory. You surely can consider axioms which allow such a thing and consider the possibilities abstractly. Also you can see some of Godel's work on trying to get really formal, yet that work pre-supposes some kind of basic set theory. Really the formal systems are just trying to make minimum statements which agree with our intuition. Also possibly see differential geometry, differential forms, also some could say... $\endgroup$ – marshal craft Feb 17 '18 at 15:03
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    $\begingroup$ Besides non-standard analysis and hyperreals, have a look at combinatorial game theory and the Surreals. $\endgroup$ – ypercubeᵀᴹ Feb 17 '18 at 23:57
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Most objections to non-standard analysis seem to be about the use of the axiom of choice in the construction of the field of hyperreals. Non-standard analysis is completely rigorous, but if you're a hardcore constructivist then you may be a bit squeamish about it. Then again, there's always some things you need to take on faith in any branch of maths:

  • If you're a hardcore finitist then you have to be really careful about analysis in general, since the conventional $\mathbb{R}$ as an object doesn't exist at all.
  • If you don't accept the axiom of dependent choices then you're pretty limited in what you can do in real analysis, because many arguments rely on taking a sequence chosen arbitrarily.
  • If you don't believe there is a nonprincipal ultrafilter on $\mathbb{N}$ then you can't construct the ultrapower required to create the hyperreals.

If you choose to allow more axioms ("there is an infinite set", "dependent choices", "there is a nonprincipal ultrafilter on $\mathbb{N}$") then you get access to correspondingly more interesting things you can do, but it's all still rigorous.

Note, however, that if you accept Choice then in a certain sense "anything you can do in non-standard analysis, you can also do without the hyperreals" (see https://math.stackexchange.com/a/51480/259262). It's an extra proof technique to make things easier by hiding many of the $\forall \exists$ quantifiers, rather than allowing you to prove genuinely new things that you couldn't prove before.

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    $\begingroup$ Again I think the question misses completely what math is about. It's not about fighting some battle on if in our physical universe, there are infinities or infinitesimals but instead, completely irrelevant to the answer to those questions, is can we consider the possibilities? Sure can. Cantor's diagonalization is valid somewhere, it may not be consistent with other areas, but you can consider it, the consequences of it, etc. The question of which one agrees with our reality, perhaps is on the edge of philosophy and mathematics, can the cannons of mathematics. $\endgroup$ – marshal craft Feb 17 '18 at 15:08
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    $\begingroup$ There are constructive and predicative approaches to creating models for nonstandard analysis. Obviously the result will be a bit different. Another approach is internal set theory which was formulated by Edward Nelson, an ultrafinitist (which is not to say everything he did or even this was ultrafinitistic). $\endgroup$ – Derek Elkins Feb 17 '18 at 19:10
  • $\begingroup$ @DerekElkins Actually, his internal set theory was inspired by his ultrafinitism. The idea was that standard number described the numbers that actually existed, and nonstandard numbers where the ones that were too large too exist. $\endgroup$ – PyRulez Feb 18 '18 at 7:58
  • $\begingroup$ Can't you use the model existence theorem to construct a model of nonstandard analysis? $\endgroup$ – PyRulez Feb 18 '18 at 7:59
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    $\begingroup$ @PyRulez OK, but you need some method for distinguishing the theorems which are true of $\mathbb{R}$ from the theorems which are true of the hyperreals. The real difficulty is in dealing with the second-order statements like Dedekind completeness. Robinson's model has a clear way of doing that, by identifying the internal sets; an arbitrary model that you grab from the model existence theorem has no clear way to get these useful correspondences of second-order theorems. (I think; it's been a while since I did this stuff.) $\endgroup$ – Patrick Stevens Feb 18 '18 at 8:51
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Your perception is wrong. Non-standard analysis is grounded on Logic and it's as solid as any other field of Mathematics. I suggest that you read Abraham Robinson's Non-standard Analysis.

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  • $\begingroup$ Given the level of the OP I would suggest Keisler's textbook Elementary Calculus instead. $\endgroup$ – Mikhail Katz Feb 18 '18 at 10:57
  • $\begingroup$ @MikhailKatz If the OP was after a textbook about infinitesimals, I would agree with you. But I think that Robinson's textbook is better to persuade someone how rigorous non-standard analysis is. $\endgroup$ – José Carlos Santos Feb 18 '18 at 11:01
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    $\begingroup$ It won't persuade him of anything if he can't follow it. $\endgroup$ – Mikhail Katz Feb 18 '18 at 12:42
  • $\begingroup$ @MikhailKatz I must admit that you make a good point there. $\endgroup$ – José Carlos Santos Feb 18 '18 at 12:44
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    $\begingroup$ Keisler is a first-rate respected logician. His textbook was published by a reputable publisher. This itself is sufficient proof that the techniques are rigorous. $\endgroup$ – Mikhail Katz Feb 18 '18 at 12:53
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Amusingly, one of the answers to the question you ask is that elementary calculus rigorously defines infinitesimals.

How does it do so? Via the notion of differential. The problem you're struggling with is almost backwards; the standard, traditional track is:

  • Define the notion of derivative
  • Use multivariable derivatives to define the notions of (tangent) vector and differential
  • Conceptualize a notion of an "infinitesimal" neighborhood of a point

To elaborate on that last point, you're supposed to envision the points of the infinitesimal neighborhood to be enumerated by tangent vectors — the intuitive idea is that you take an "infinitesimal" step proportional to the tangent vector. Differentials are the functions on the infinitesimal neighborhood.

But this conceptualization is not trying to define anything new — it is merely a way of thinking about calculus. (albeit a very useful one!)

But the second point is bog standard. For example, in multivariable calculus, one incarnation of these notions is

  • Tangent vectors to points in $\mathbb{R}^n$ are $n \times 1$ column vectors — the sort of thing you get when differentiating a vector function of one variable
  • Differentials at points in $\mathbb{R}^n$ are $1 \times n$ row vectors — the sort of thing you get when differentiating a scalar function of $n$ variables

Both the subjects of differential geometry and algebraic geometry treat this sort of thing much more explicitly and more in-depth.

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To answer the title question: "Which branch of mathematics rigorously defines infinitesimals?": the answer is "a serious undergraduate course in algebra, including the existence of a maximal ideal". Indeed, that is all that's required to construct a hyperreal extension of the real field satisfying the requisite properties, such as existence of infinitesimals, etc.

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Like others have pointed out, nonstandard analysis is perfectly rigorous, just a limited more complicated to set up. However, I will list some other, equally rigorous approaches to infinitesimals.

  • Dual numbers: These are very simple to set up. A dual number is basically just a ordered pair of numbers. It is based on an infinitesimal called $\epsilon$, which is a number such that $\epsilon^2 = 0$. Their relation to differentiation is $f(a + b\epsilon) = f(a) + bf'(a)\epsilon$. In fact, if you use a programming language that implements dual numbers (like Haskell), you can find the derivative of any function by solving that instead of approximating the derivative with a secant line. However, do not expect to get very far in Calculus with them: they only help you define derivatives, they have no transfer principle, and they are not even a field.
  • Surreal numbers: This fun little number system includes real numbers, infinitesimals, cardinal number, ordinal numbers, and every ordered field. It is also satisfies the field axioms itself. However, they have nearly no applications in Calculus, because they are too different from real numbers. They were actually invented for use in game theory, their seminal text being On Numbers and Games (which is a great book, by the way). They also have an extremely simple definition given their complexity, even simpler than the real numbers. Conway actually proposes that it would be simpler to construct the real numbers by first constructing the surreal numbers, and then getting rid of all the non-real surreal numbers.
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There's an approach, by the names of synthetic differential geometry or smooth infinitesimal analysis, which considers a kind of 'real line extended with' nilpotent elements, in this case, objects $ d $ not necessarily equal to $ 0 $ satisfying

$ d^2 = 0 $

Note this implies one's no longer working with/inside a conventional algebraic field, and indeed the axiomatic development of this theory, though in some sense simpler than the more usual forms of Analysis, demands some specially careful attention

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