A Prove that points which are indistinguishable from 0 exist (using logic)

[Mike_bb]

Hello!

I read about Smooth infinitesimal analysis (https://www.humanities.mcmaster.ca/~rarthur/papers/LsiSiaNp6.rev.pdf, page 24) (Synthetic differential geometry) and I encountered a problem.

Some authors introduce such thing as "infinitesimal neighborhood" (but they don't prove its existence, they introduce it as definition). For example:

Bell explains this as follows. Define two points $a$ and $b$ on the real line (as represented in a smooth world $S$) as
distinguishable iff they are not identical, i.e. iff $\lnot a = b$, where ‘=’ denotes identity. Now define the "infinitesimal neighbourhood" $I(0)$ of a given point 0 as the set of all those points indistinguishable from 0. That is, define $I(0)$ as follows:

$I(0) = \{x | \lnot \lnot x=0 \}$

Now if the Law of Double Negation (or, equivalently, the Law of Excluded Middle) held in $S$, we could infer that $x = 0$ for each $x$ in $I(0)$, so that the infinitesimal neighbourhood of $0 I(0)$ would reduce to $\{ 0 \}$. But we know that this neighbourhood does not reduce to $\{ 0 \}$ in S. So we cannot infer the identity of points from their inddistinguishability. Again, suppose there is a point $a$ in $I$ that is distinguishable from 0, i.e. suppose there is a point $a$ in $I$ such that $\lnot a = 0$.But since $a$ in $I$, $\lnot \lnot a = 0$ by definition. But this is a contradiction. Therefore it is not the case that there exists a point $a$ in $I$ that is distinguishable from 0.

My question is: How to prove using (intuitionistic) logic that in "infinitesimal neighborhood" there are points that indistinguishable from 0 and not identical to 0?

Thanks!

 

[jedishrfu]

Is this related to the notion of hyperreal numbers?

https://en.wikipedia.org/wiki/Hyperreal_number

 

[Hornbein]

My question is: How to prove using (intuitionistic) logic that in "infinitesimal neighborhood" there are points that indistinguishable from 0 and not identical to 0?

Thanks!

I'm no expert but feel pretty sure this can't be proved constructively. It has to do with the continuum hypothesis. It was proved long ago that math is consistent whether or not one accepts the continuum hypothesis. If you accept it then no such thing can occur, if you reject it then you can assume such a thing exists without introducing any contradictions.

 

[Dale]

I don’t think it is correct either. It seems like they are assuming the existence of this neighborhood.

 

[FactChecker]

How can you say that two points are indistinguishable and yet not identical? When you prove they are not identical, doesn't that prove they are distinguishable?

 

[Mike_bb]

Is this related to the notion of hyperreal numbers?

https://en.wikipedia.org/wiki/Hyperreal_number

No. This related to: https://en.wikipedia.org/wiki/Smooth_infinitesimal_analysis

How can you say that two points are indistinguishable and yet not identical? When you prove they are not identical, doesn't that prove they are distinguishable?

It's written in John Bell's book "A Primer of Infinitesimal Analysis":

 

[FactChecker]

No. This related to: https://en.wikipedia.org/wiki/Smooth_infinitesimal_analysis


It's written in John Bell's book "A Primer of Infinitesimal Analysis":

View attachment 358581

That confuses me more. It defines "distinguishable" as "not identical" That seems to support my objection.
I will have to leave this for others who know more about infinitesimal analysis. I know nothing about it.

 

[jedishrfu]

Perhaps @fresh_42 can provide some sense of clarity here.

 

[fresh_42]

Perhaps @fresh_42 can provide some sense of clarity here.

Infinitesimals + history + philosophy + logic + common language + page 24 + a book I do not possess:
This is a mine field.

One has to be very exact here, not to step on a mine, i.e., confuse terms that might be or have been used differently in time, in both sources, in common language. I could possibly add even another perspective, but I do not see how such a technical question can be decided without access to all sources (Leibniz, Bell, Arthur).

 

[jedishrfu]

Thanks @fresh_42

 

[fresh_42]

It looks like they want to express that there are more than one infinitesimal and that they are not zero. Distinguishability is a bit complicated since we have two of them (or more) with the same behavior and name, yet not identical. The picture on Wikipedia tries to shed light onto that apparent paradox:

However, I'm not 100% sure whether this is actually what is all about without reading the sources.

 

[Klystron]

However, I'm not 100% sure whether this is actually what is all about without reading the sources.

I read partway through the philosophy paper in the OP. Appears rather typical of an updated "how many angels can dance on the point of a pin" question popular with medieval scholars, with some notable lacunae. Many references to Leibniz and Newton but only one mention of Cantor in a title for a set of points. Author jumps to Bell yet no mention of Planck. Likely I am missing the point (NPI).

Using obsolete terminology such as syncategorematic tends to obfuscate rather than clarify.

 

[fresh_42]

I read partway through the philosophy paper in the OP. Appears rather typical of an updated "how many angels can dance on the point of a pin" question popular with medieval scholars, with some notable lacunae. Many references to Leibniz and Newton but only one mention of Cantor in a title for a set of points. Author jumps to Bell yet no mention of Planck. Likely I am missing the point (NPI).

Using obsolete terminology such as syncategorematic tends to obfuscate rather than clarify.

It is not easy. We have $c+\varepsilon=c+2\varepsilon$ for real $c$ but at the same time $\varepsilon\neq 0$ and obviously $\varepsilon\neq 2\varepsilon.$

Now try to find a logically rigorous wording using common language. This isn't an easy task and I doubt that Leibinz or philosophy are of any help here. Popper's excursion into science wasn't what I would call a success. At least Popper had a very prominent critic, Einstein. There was much intuition in the game at Leibniz's or Newton's time. And I'm not even sure which John Bell is meant. It is quite a common name. I hope it's not the physicist John Stewart Bell and the mathematician John B. Bell instead. However, the first line I read was "Bell was an American applied mathematician." What? Nowhere any logician in sight? That's a strange ensemble of people discussing a concept that definitely requires precise logic.

 

[martinbn]

It looks like a discussion about how many infinitesimals can fit on the head of a pin.

The paper you linked to is by someone in a philosophy department, and the paper itself seems to be about history and philosophy.

 

[pbuk]

This:

The paper you linked to is by someone in a philosophy department, and the paper itself seems to be about history and philosophy.

But in any case this question:

My question is: How to prove using (intuitionistic) logic that in "infinitesimal neighborhood" there are points that indistinguishable from 0 and not identical to 0?

is already answered in your OP:

Some authors introduce such thing as "infinitesimal neighborhood" (but they don't prove its existence, they introduce it as definition). For example:

This is how the mathematics of number systems works - we define a set, then we add any necessary axioms, then we prove some things and see if what we have is consistent, and then we have a number system. Whether the elements of the set "exist" or not is completetly irrelevent to mathematics - it is a question for philosophy.

What we can say with certainty in the context of the infinitessimals is that the elements of the infinitesimal neighbourhood do not exist within the reals, because that is how we have defined them.

If you have difficulty accepting this consider instead the complex numbers. We start by defining $i: i^2 = -1$ and we then open up a whole world of mathemetics with many practical applications, however at no point do we stop to consider whether any $a + ib$ actually exists. Of course if we were to do so then we would first have to define what we mean when we say something "exists", and then we are no longer mathematicians but have become philosophers and must find another forum to continue our exploration.

Edit: Die ganzen Zahlen hat der liebe Gott gemacht, alles andere ist Menschenwerk

 

[fresh_42]

Edit: Die ganzen Zahlen hat der liebe Gott gemacht, alles andere ist Menschenwerk

Leopold Kronecker, quoted in F. Cajori's A History of Mathematics (1919).

 

[pbuk]

Leopold Kronecker, quoted in F. Cajori's A History of Mathematics (1919).

I did not realise that you had already referenced the same quotation recently

 

[fresh_42]

I did not realise that you had already referenced the same quotation recently

I kind of like Kronecker which is why I added it. He was an interesting character. I once stumbled upon an article in a journal in which he commented, better 'destroyed', someone else's paper. I'm glad he cannot read my insight articles. I don't even agree with his statement about integers. I consider myself a Platonist and think of mathematical concepts as existing ideas from which we only have a real-world projection to work with.