taeto

I am curious about the assumption of such a function \(f.\) Does it provably exist in such a theory (presumably using second order logic) in which there are infinitesimals? Or is the existence undecidable? I have little experience about the possibility of undecidable statements in second order theories. As a motivation for my question, if I formulate the "in other words" condition for an infinitesimal \(x\) as \[ x \neq 0 \mbox{ and } \forall n\in \mathbb{N} \,:\, n\cdot |x| < 1, \] and if we assume that \(\mathbb{N}\) is the usual and not necessarily "standard" version of the natural numbers, then there are models in which the \(n\) can be infinite. And in that event, it intuitively seems a very strong condition on an \(x \neq 0\) to have \(n\cdot |x| < 1.\) I am insufficiently familiar with second order theories to know whether the theory can possible "see" (express formally) that a natural number \(n\) is actually finite. If so, then maybe the "in other words" condition needs added assumptions, such as the finiteness of \(n?\)

Thanks for the useful link! Regarding an "original theory", such a one would have a theorem like \( dy = \frac{dy}{dx}dx,\) which means that you can do arithmetic with an "infinitesimal" \(dx\), assumed nonzero. In contemporary mathematics the same expression is still a theorem, but it stands for something entirely different; both \(x\) and \(y\) are functions that have differentials \(dx\) and \(dy\) respectively, with \(\frac{dy}{dx}\) being their derivative. None of the latter functions represent anything "infinitely small", indeed the range of either differential can easily be unbounded. In that sense the "theories" somehow should not be considered comparable, because they speak of completely different things. On the other hand, they might be, at least partially, "isomorphic", by being able to show theorems that have an identical outward appearance. I am definitely interested in any small nuances which would make a proposed "original theory" make a different prediction than what we would expect today.

I have read somewhere that Cauchy tried to use the "original theory" in an argument, but ended up with a wrong result. I will look for the reference, though maybe someone knows already?!

But then to you, technology would mean how better to swing between the tree branches, while to the rest of us, science means to ascertain what is out there and how to adapt the environment to suit our lives.

Okay. Teach me.

You happen to possess the historical reference for a time when the state of affairs of science was "good", meaning that everything was already understood?

blockquote widget An obvious reason might be: when I show you a definition that is not circular, you immediately complain that it is circular. That tells me quite a bit.

I know what it means, But I am getting pretty convinced that you do not. That is why I ask.

Explain what you mean by "circular", please.

We can explain what a second is to you. If you are holding in your hand, while managing not to shake it very much, a caesium 133 atom which you observe being in its ground state, and you count 9 192 631 770 periods of the radiation corresponding to the transition between the two hyperfine levels of this ground state, then you have just experienced one second of time passed. Moreover, anybody or anything that stayed fixed in space with respect to you during the entire counting process has experienced exactly one second as well. Now just as you can measure the distance between different things using a meter stick, you can measure the time passed between different events using this device, which is commonly called a "clock".

A forum for "Theoretical Computer Science" would be great for me personally. The current "Computer Science" forum is more like a "Computer Repairs" (not wanting to be facetious, though maybe a little bit) item.