  # dasnulium

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1. wtf: I did think carefully about word choice for the thesis, in particular I would never use the term 'infinitely small' to describe an infinitesimal - 'indefinitely small' does not mean smaller than 'every' positive number, it means smaller than any positive number to which you can assign a value (this is similar to Kant's ideas about the infinite as discussed by Bell). I don't use the word 'arbitrary' because in numerical analysis (unlike in regular calculus) it means something different to 'indefinite' - namely that the minimum value of the increment may be arbitrary (alternatively, depending on the functions concerned, it may have to meet certain criteria). Note that the term 'indefinitely small' is meaningless for numerical analysis for obvious reasons. uncool: Bell's take on the 'blip' function doesn't distinguish between non-zero and zero/infinitesimal, but simply between non-zero and zero - so I don't see the purpose of your question, maybe my answer to wtf would help.
2. The point of the paper is that there's a connection that may have been overlooked, maybe for this reason: He's ruling out the idea that the standard part/nilsquare rule operation is somehow unsafe. A limit in calculus is what you get when the increment of x becomes indefinitely small (i.e. infinitesimal) so I think that they are two ways of looking at the same thing. You also mention statistics, but I know that there have been efforts to found statistics on a non-constructive basis, which I would avoid so I can't comment on that. Replying to uncool: the function you describe is explicitly discontinuous so I would simply assume calculus doesn't apply to it. Bell talks about it on page 5 here https://pdfs.semanticscholar.org/e226/af69111bcba4aff8318f2b479dd6c3202325.pdf Clarification to last comment: the equation s = h √(1 + y'²) applies to the finite difference and is therefore true for any value (in this context 'proportional' is the wrong word for the RHS). Does it apply to infinitesimal increments? I said that "Note that SIA seems to work better than NSA for this", but you can get it to work in NSA too. First do this s/h = √(1 + y'²), then transition to the infinitesimal by changing s/h to ds/dh (yielding the well known equation) and taking the standard part of y'. This may be taken to mean that ds/dh becomes 0/0, but it couldn't subsequently be neglected because it would be indeterminate, not zero.
3. To elaborate on "the 'error' for polynomials is the sum of the higher power incremental terms" - when we take the derivative of y = f(x), x is said to be the independent variable. However, differentiation assumes an arbitrary x value and a variable increment, and the latter is the quantity actually varying. If we take the finite difference quotient of a polynomial, varying the increment (for a given x value) results in a gradient that changes with the secant; but if we take the regular derivative of that polynomial we can't vary the increment because those terms have been neglected (and cancelled) - and since an indefinitely small increment implies an indefinitely small secant length*, which is by definition part of the tangent, this gives us the gradient of the tangent. *By Pythagoras secant length is proportional to the increment thus s = h √(1 + y'²) although the equation is not linear, so simply by reducing the increment a smaller secant length can be found. The derivative may of course increase to counteract this, but it would have to eventually become vertical to nullify it, at which point calculus no longer applies. Note that SIA seems to work better than NSA for this line of reasoning because taking the standard part dispenses with all incremental terms, even those of the first power, but this would cause the first RHS term to be set to zero. This may indicate that SIA bears a closer resemblance to the true nature of calculus than NSA.
4. "How is LEM or its denial a corollary of nilpotency?" Because an increment which produces a noticeable 'error' (defined relatively) is by definition not infinitesimal and therefore LEM applies. The 'error' for polynomials is the sum of the higher power incremental terms. LEM here means non-negligible, not separate in some ideal sense (i.e. axiomatically). That's why I question the justification for LEM in the paper. I can't talk about the background further because the main point of the paper is to show that limits and original infinitesimals are technically equivalent at least, which is very simple. As Klein said "With these elementary presentations there remains always the suspicion that by neglecting successive small quantities we have finally accumulated a noticeable error, even if each is permissable singly. But I do not need to show how we can rule this out, for these questions are so thoroughly elementary that each of you can think them through when you feel so inclined." Elementary Mathematics - From an Advanced Standpoint, Felix Klein, 1908, p190 (NB that is a paraphrased translation). Very few people ever do feel so inclined though. For more of the philosophical background The Continuous and the Infinitesimal by John L Bell is the best guide. Over and out!
5. That was the part where I try to discern why a foundational crisis even happened in mathematics which is not the main point of the paper, so I won't get into a debate about that. But you might like this guy's take on it: https://www.quora.com/profile/Eric-Freiling.
6. wtf: Could you quote the line from the paper which is 'demonstrably false'.
7. "You will never come up with a well-order [sic] of the reals and neither will anyone else. It's a consequence of the Axiom of Choice (AC), so it is inherently nonconstructive." wtf
8. wtf: Yes, it is related to SIA - one of my main influences was John L Bell. I was very annoyed after reading one of his books because the simpler proofs offered by SIA which he uses had not been available when I was at school. Of course, they're only simpler for polynomials, but that's an important class of functions. Another very important class of functions is the mechanical functions, which Descartes excludes from consideration in a quote in the paper. But even if they can't be analyzed normally they can be analyzed numerically i.e. with finite differences, and if you take Granville's approach (as in the last reference) as the best example of how limits can be applied then finite differences and limits work in a very similar way. This is of course much easier with computers - for a tangent you get a list of numbers converging on a value. Since this approach does also work for polynomials we can say that limits are a more general theory - Leibniz implies this in a quote I give where he gives what to me seems like the 19th century limit definition. Note that before computers were invented there wasn't much incentive to think about the broader theory and angst about the continuum didn't 'boil over' until two hundred years after Leibniz and Newton. Bell has written much about the continuum - constructivism holds that saying something is not unequal to something (e.g. zero) does not imply that it is actually equal to it. In pure mathematics where we can just give a variable a value (as opposed to real life) the only way that that condition can be met is if something is smaller than any value you give - which is the essence of the limit criterion and also a description of indefinite smallness, which is why I think they're the same thing and say that LEM has been over-applied. Anyway, when I said Leibniz provoked more debate I meant constructive debate - although not by much since Leibniz and Nieuwentijt never reconciled their methods. If they had I probably wouldn't be writing this, but my job requires very obsessive thinking so the paper was a natural development.