Infinitesimals

[Philip Thrift]

Leibniz's Infinitesimals: Their Fictionality, Their Modern Implementations, And Their Foes From Berkeley To Russell And Beyond

https://arxiv.org/abs/1205.0174

Infinitesimals, Imaginaries, Ideals, and Fictions

https://arxiv.org/abs/1304.2137

Leibniz vs Ishiguro: Closing a quarter-century of syncategoremania

https://arxiv.org/abs/1603.07209

Leibniz frequently writes that his infinitesimals are useful fictions, and we agree; but we shall show that it is best not to understand them as logical fictions; instead, they are better understood as pure fictions.

@philipthrift

[Lawrence Crowell]

We can think of infinitesimals as a manifestation of Gödel's theorem with Peano number theory. There is nothing odd that is going to happen with this number theory, but no matter how much we count we never reach "infinity." We have then an issue of ω-consistency, and to completeness. To make this complete we must then say there exists an element that has no successor. We can now take this "supernatural number" and take the reciprocal of it within the field of rationals or reals. This is in a way what infinitesimals are. These are a way that Robinson numbers are constructed. These are as "real" in a sense, just as imaginary numbers are. They are only pure fictions if one stays strictly within the Peano number theory. They also have incredible utility in that the whole topological set theory foundation for algebraic geometry and topology is based on this.

LC

[Bruno Marchal]

On 9 Nov 2019, at 02:22, Lawrence Crowell <goldenfield...@gmail.com> wrote:

We can think of infinitesimals as a manifestation of Gödel's theorem with Peano number theory. There is nothing odd that is going to happen with this number theory, but no matter how much we count we never reach "infinity." We have then an issue of ω-consistency, and to completeness. To make this complete we must then say there exists an element that has no successor. We can now take this "supernatural number" and take the reciprocal of it within the field of rationals or reals. This is in a way what infinitesimals are. These are a way that Robinson numbers are constructed. These are as "real" in a sense, just as imaginary numbers are. They are only pure fictions if one stays strictly within the Peano number theory. They also have incredible utility in that the whole topological set theory foundation for algebraic geometry and topology is based on this.

Roughly thinking, I agree. It corroborates my feeling that first order logic is science, and second-order logic is philosophy. Useful philosophy, note, but useful fiction also.

Bruno

LC

On Sunday, November 3, 2019 at 6:39:53 AM UTC-6, Philip Thrift wrote:

Leibniz's Infinitesimals: Their Fictionality, Their Modern Implementations, And Their Foes From Berkeley To Russell And Beyond

https://arxiv.org/abs/1205.0174

Infinitesimals, Imaginaries, Ideals, and Fictions

https://arxiv.org/abs/1304.2137

Leibniz vs Ishiguro: Closing a quarter-century of syncategoremania

https://arxiv.org/abs/1603.07209

Leibniz frequently writes that his infinitesimals are useful fictions, and we agree; but we shall show that it is best not to understand them as logical fictions; instead, they are better understood as pure fictions.

@philipthrift

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[Lawrence Crowell]

On Sunday, November 10, 2019 at 6:17:10 AM UTC-6, Bruno Marchal wrote:

On 9 Nov 2019, at 02:22, Lawrence Crowell <goldenfield...@gmail.com> wrote:

We can think of infinitesimals as a manifestation of Gödel's theorem with Peano number theory. There is nothing odd that is going to happen with this number theory, but no matter how much we count we never reach "infinity." We have then an issue of ω-consistency, and to completeness. To make this complete we must then say there exists an element that has no successor. We can now take this "supernatural number" and take the reciprocal of it within the field of rationals or reals. This is in a way what infinitesimals are. These are a way that Robinson numbers are constructed. These are as "real" in a sense, just as imaginary numbers are. They are only pure fictions if one stays strictly within the Peano number theory. They also have incredible utility in that the whole topological set theory foundation for algebraic geometry and topology is based on this.

Roughly thinking, I agree. It corroborates my feeling that first order logic is science, and second-order logic is philosophy. Useful philosophy, note, but useful fiction also.

Bruno

The key word is useful. Infinitesimals are immensely useful in calculus and point-set topology. It provide a proof of the mean value theorem in calculus, which in higher dimension is Stokes' rule that in the language of forms lends itself to algebraic topology. Something that useful as I see it has some sort of ontology to it, even if it is in the abstract sense of mathematics.

LC

 

LC

On Sunday, November 3, 2019 at 6:39:53 AM UTC-6, Philip Thrift wrote:

Leibniz's Infinitesimals: Their Fictionality, Their Modern Implementations, And Their Foes From Berkeley To Russell And Beyond

https://arxiv.org/abs/1205.0174

Infinitesimals, Imaginaries, Ideals, and Fictions

https://arxiv.org/abs/1304.2137

Leibniz vs Ishiguro: Closing a quarter-century of syncategoremania

https://arxiv.org/abs/1603.07209

Leibniz frequently writes that his infinitesimals are useful fictions, and we agree; but we shall show that it is best not to understand them as logical fictions; instead, they are better understood as pure fictions.

@philipthrift

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[Philip Thrift]

On Sunday, November 10, 2019 at 1:09:41 PM UTC-6, Lawrence Crowell wrote:

On Sunday, November 10, 2019 at 6:17:10 AM UTC-6, Bruno Marchal wrote:

On 9 Nov 2019, at 02:22, Lawrence Crowell <goldenfield...@gmail.com> wrote:

We can think of infinitesimals as a manifestation of Gödel's theorem with Peano number theory. There is nothing odd that is going to happen with this number theory, but no matter how much we count we never reach "infinity." We have then an issue of ω-consistency, and to completeness. To make this complete we must then say there exists an element that has no successor. We can now take this "supernatural number" and take the reciprocal of it within the field of rationals or reals. This is in a way what infinitesimals are. These are a way that Robinson numbers are constructed. These are as "real" in a sense, just as imaginary numbers are. They are only pure fictions if one stays strictly within the Peano number theory. They also have incredible utility in that the whole topological set theory foundation for algebraic geometry and topology is based on this.

Roughly thinking, I agree. It corroborates my feeling that first order logic is science, and second-order logic is philosophy. Useful philosophy, note, but useful fiction also.

Bruno

The key word is useful. Infinitesimals are immensely useful in calculus and point-set topology. It provide a proof of the mean value theorem in calculus, which in higher dimension is Stokes' rule that in the language of forms lends itself to algebraic topology. Something that useful as I see it has some sort of ontology to it, even if it is in the abstract sense of mathematics.

LC

 

 

It is interesting that infinitesimal calculus [ https://www.math.wisc.edu/~keisler/foundations.pdf ] is still a "backbench" calculus - not mattering so much in science, at least in terms of education. Maybe that's a problem with science.

@philipthrift

[Bruno Marchal]

On 10 Nov 2019, at 20:09, Lawrence Crowell <goldenfield...@gmail.com> wrote:

On Sunday, November 10, 2019 at 6:17:10 AM UTC-6, Bruno Marchal wrote:

On 9 Nov 2019, at 02:22, Lawrence Crowell <goldenfield...@gmail.com> wrote:

We can think of infinitesimals as a manifestation of Gödel's theorem with Peano number theory. There is nothing odd that is going to happen with this number theory, but no matter how much we count we never reach "infinity." We have then an issue of ω-consistency, and to completeness. To make this complete we must then say there exists an element that has no successor. We can now take this "supernatural number" and take the reciprocal of it within the field of rationals or reals. This is in a way what infinitesimals are. These are a way that Robinson numbers are constructed. These are as "real" in a sense, just as imaginary numbers are. They are only pure fictions if one stays strictly within the Peano number theory. They also have incredible utility in that the whole topological set theory foundation for algebraic geometry and topology is based on this.

Roughly thinking, I agree. It corroborates my feeling that first order logic is science, and second-order logic is philosophy. Useful philosophy, note, but useful fiction also.

Bruno

The key word is useful. Infinitesimals are immensely useful in calculus and point-set topology.

Which infinitesimals? The informal one by Newton or Leibniz? Their recovering in non-standard analysis?

Of in synthetic (category based) geometry?

Personally, despite I am logician, I don’t really believe in non standard analysis. I find the Cauchy sequences more useful, and directly understandable (the “new” infinitesimal requires an appendix in either mathematical logic or in category theory).

It provide a proof of the mean value theorem in calculus, which in higher dimension is Stokes' rule that in the language of forms lends itself to algebraic topology.

Abstract topology is enough here, in the Kolmogorov topological abstract spaces. You don’t need formal infinitesimal to have a mean value theorem in calculus. I guess you are OK with this.

Something that useful as I see it has some sort of ontology to it, even if it is in the abstract sense of mathematics.

Like physics, when we assume mechanism, it exists in the phenomenological sense, which is the case of all interesting thing. But to solve the mind-body problem, we need to be clear on the ontology, and with mechanism, the natural numbers (accompanied by their usual + and * laws) or anything Turing equivalent is enough, and cannot be extended, without making the phenomenology exploding (full of “white rabbits”).

Bruno

LC

 

LC

On Sunday, November 3, 2019 at 6:39:53 AM UTC-6, Philip Thrift wrote:

Leibniz's Infinitesimals: Their Fictionality, Their Modern Implementations, And Their Foes From Berkeley To Russell And Beyond

https://arxiv.org/abs/1205.0174

Infinitesimals, Imaginaries, Ideals, and Fictions

https://arxiv.org/abs/1304.2137

Leibniz vs Ishiguro: Closing a quarter-century of syncategoremania

https://arxiv.org/abs/1603.07209

Leibniz frequently writes that his infinitesimals are useful fictions, and we agree; but we shall show that it is best not to understand them as logical fictions; instead, they are better understood as pure fictions.

@philipthrift

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[Philip Thrift]

On Monday, November 11, 2019 at 3:28:05 AM UTC-6, Bruno Marchal wrote:

Personally, despite I am logician, I don’t really believe in non standard analysis. I find the Cauchy sequences more useful, and directly understandable (the “new” infinitesimal requires an appendix in either mathematical logic or in category theory).

But, what about Zeno? :)

Actualized limits? Can't happen in physical reality.

@philipthrift 

[Lawrence Crowell]

On Monday, November 11, 2019 at 3:28:05 AM UTC-6, Bruno Marchal wrote:

On 10 Nov 2019, at 20:09, Lawrence Crowell <goldenfield...@gmail.com> wrote:

On Sunday, November 10, 2019 at 6:17:10 AM UTC-6, Bruno Marchal wrote:

On 9 Nov 2019, at 02:22, Lawrence Crowell <goldenfield...@gmail.com> wrote:

We can think of infinitesimals as a manifestation of Gödel's theorem with Peano number theory. There is nothing odd that is going to happen with this number theory, but no matter how much we count we never reach "infinity." We have then an issue of ω-consistency, and to completeness. To make this complete we must then say there exists an element that has no successor. We can now take this "supernatural number" and take the reciprocal of it within the field of rationals or reals. This is in a way what infinitesimals are. These are a way that Robinson numbers are constructed. These are as "real" in a sense, just as imaginary numbers are. They are only pure fictions if one stays strictly within the Peano number theory. They also have incredible utility in that the whole topological set theory foundation for algebraic geometry and topology is based on this.

Roughly thinking, I agree. It corroborates my feeling that first order logic is science, and second-order logic is philosophy. Useful philosophy, note, but useful fiction also.

Bruno

The key word is useful. Infinitesimals are immensely useful in calculus and point-set topology.

Which infinitesimals? The informal one by Newton or Leibniz? Their recovering in non-standard analysis?

Of in synthetic (category based) geometry?

If one is sticking to a more formal approach then Leibniz  Really Weierstrass is the guy who got this straight. 

 

Personally, despite I am logician, I don’t really believe in non standard analysis. I find the Cauchy sequences more useful, and directly understandable (the “new” infinitesimal requires an appendix in either mathematical logic or in category theory).

These things are not about belief or nonbelief. They are formal models, and as I see it one works with any particular model if it is useful. 

 

It provide a proof of the mean value theorem in calculus, which in higher dimension is Stokes' rule that in the language of forms lends itself to algebraic topology.

Abstract topology is enough here, in the Kolmogorov topological abstract spaces. You don’t need formal infinitesimal to have a mean value theorem in calculus. I guess you are OK with this.

The MVT relies upon calculus f'(c) = (f(a) - f(b))/(a - b) or the integral form ∫f(x)dx = f(c)(a - b) for b to a limits in integral. So infinitesimals are there at least implicitly. 

When it comes to point set topology I prefer to get past that as quickly as possible and get to cohomology, homotopy or cobordism.

 

Something that useful as I see it has some sort of ontology to it, even if it is in the abstract sense of mathematics.

Like physics, when we assume mechanism, it exists in the phenomenological sense, which is the case of all interesting thing. But to solve the mind-body problem, we need to be clear on the ontology, and with mechanism, the natural numbers (accompanied by their usual + and * laws) or anything Turing equivalent is enough, and cannot be extended, without making the phenomenology exploding (full of “white rabbits”).

Bruno

I don't have thoughts on the mind-body problem. I have no particular theory about consciousness or anything related.

LC