(I’ve probably posted all of this before, but may as well reiterate as my opinions have shifted slightly since the olden days).

I’ve put this viewpoint on a multipart youtube. Part 1 is https://www.youtube.com/watch?v=s9OVj_XmvTY see links in the comment for the other parts. The part with least mathematics is https://www.youtube.com/watch?v=M8TwodhqRoM

In primary school we learn that ∞ = ∞+1, that ∞/∞ is undefined, and that ∞*0 is undefined. Later we learn that infinite series sometimes converge and sometimes diverge, and that -∞ is not a number. And in university learn that log(∞) sometimes exists and sometimes doesn’t exist and when it does exist then log(∞) < ∞, even though sqrt(∞) = ∞. This version of infinity was invented by Cantor who was totally mad, was championed by Peano who was half mad, and was ruthlessly promoted by Hilbert who went mad in his old age. This standard version of ∞ is useless. There is an alternative.

You may have already used nonstandard analysis without knowing it. If you write -∞ on your graph, or dy/dx for a differential, or use the order of magnitude symbol O(), then you’re already using nonstandard analysis. I’m going to use ω for infinity in nonstandard analysis to avoid confusion. Nonstandard analysis has been rigorously proved to be derivable in four different ways. From wikipedia.org/wiki/Transfer_principle, from Surreal numbers, from Hyperreal numbers, and from Hahn series. The Transfer principle can be taught in primary school and the Surreal numbers in high school.

There is already one textbook for teaching calculus using nonstandard analysis, but it’s not a particularly good textbook.

The Transfer principle is “If something is true for all sufficiently large x then it is taken to be true for infinity ω”. Because x-1 < x < x+1 so ω-1 < ω < ω, and ω/ω = 1, and ω*0 = 0, and –ω is a number. Infinitesimals such as dy and dx exist. And, with the correct definition of limit, infinite series always converge. Infinity finally makes sense, and having ω/ω = 1 allows us to use renormalisation in quantum mechanics without the need for an ultraviolet cut-off.

Many mathematicians foam at the mouth if I suggest any of these. There is a well-known proof that ω is the smallest infinite number contradicting ω-1 < ω. When examined in detail, this proof turns out to be a tautology – you can prove that ω is the smallest infinite number if and only if you assume that ω is the smallest infinite number. Cantor published three proofs that infinitesimals can’t exist, all three proofs have since been demolished. The famous “Hilbert’s hotel” has multiple alternative interpretations. https://www.youtube.com/watch?v=Rziki9WEdRE

Many famous mathematicians have contributed to the theory of nonstandard analysis, including Newton, Leibniz, Cauchy, Laurent, Levi-Civita, the young Hilbert (before he went mad), (Rumanajan’s) Hardy, Borel, John Horton Conway, and many other mathematicians who are less well known.

Throw standard analysis in the waste bin and teach nonstandard analysis in school instead. By the way, standard analysis can be recovered from nonstandard analysis using an equivalence class.

mollwollfumble said:

(I’ve probably posted all of this before, but may as well reiterate as my opinions have shifted slightly since the olden days).

I’ve put this viewpoint on a multipart youtube. Part 1 is https://www.youtube.com/watch?v=s9OVj_XmvTY see links in the comment for the other parts. The part with least mathematics is https://www.youtube.com/watch?v=M8TwodhqRoM

In primary school we learn that ∞ = ∞+1, that ∞/∞ is undefined, and that ∞*0 is undefined. Later we learn that infinite series sometimes converge and sometimes diverge, and that -∞ is not a number. And in university learn that log(∞) sometimes exists and sometimes doesn’t exist and when it does exist then log(∞) < ∞, even though sqrt(∞) = ∞. This version of infinity was invented by Cantor who was totally mad, was championed by Peano who was half mad, and was ruthlessly promoted by Hilbert who went mad in his old age. This standard version of ∞ is useless. There is an alternative.

You may have already used nonstandard analysis without knowing it. If you write -∞ on your graph, or dy/dx for a differential, or use the order of magnitude symbol O(), then you’re already using nonstandard analysis. I’m going to use ω for infinity in nonstandard analysis to avoid confusion. Nonstandard analysis has been rigorously proved to be derivable in four different ways. From wikipedia.org/wiki/Transfer_principle, from Surreal numbers, from Hyperreal numbers, and from Hahn series. The Transfer principle can be taught in primary school and the Surreal numbers in high school.

There is already one textbook for teaching calculus using nonstandard analysis, but it’s not a particularly good textbook.

The Transfer principle is “If something is true for all sufficiently large x then it is taken to be true for infinity ω”. Because x-1 < x < x+1 so ω-1 < ω < ω, and ω/ω = 1, and ω*0 = 0, and –ω is a number. Infinitesimals such as dy and dx exist. And, with the correct definition of limit, infinite series always converge. Infinity finally makes sense, and having ω/ω = 1 allows us to use renormalisation in quantum mechanics without the need for an ultraviolet cut-off.

Many mathematicians foam at the mouth if I suggest any of these. There is a well-known proof that ω is the smallest infinite number contradicting ω-1 < ω. When examined in detail, this proof turns out to be a tautology – you can prove that ω is the smallest infinite number if and only if you assume that ω is the smallest infinite number. Cantor published three proofs that infinitesimals can’t exist, all three proofs have since been demolished. The famous “Hilbert’s hotel” has multiple alternative interpretations. https://www.youtube.com/watch?v=Rziki9WEdRE

Many famous mathematicians have contributed to the theory of nonstandard analysis, including Newton, Leibniz, Cauchy, Laurent, Levi-Civita, the young Hilbert (before he went mad), (Rumanajan’s) Hardy, Borel, John Horton Conway, and many other mathematicians who are less well known.

Throw standard analysis in the waste bin and teach nonstandard analysis in school instead. By the way, standard analysis can be recovered from nonstandard analysis using an equivalence class.

If you have said all that before, it hasn’t been as clear and concise as above, or maybe I’ve just forgotten it.

I had to look up “ultraviolet cut-off”, which shows how much I know, but anyway:

Why the desire to avoid cut-offs when applying maths to real things? Mathematical models are just simplifications and approximations to what really happens, so at some stage it becomes pointless to increase the precision of the calculation, because it isn’t increasing the accuracy.

IMO arbitrary cut-offs should be embraced as a recognition that we don’t know what is happening at that level, which would help avoid such nonsense as extrapolating to singularities, then pretending that the singularity is or was a real thing.

The Rev Dodgson said:

If you have said all that before, it hasn’t been as clear and concise as above, or maybe I’ve just forgotten it.

I had to look up “ultraviolet cut-off”, which shows how much I know, but anyway:

Why the desire to avoid cut-offs when applying maths to real things? Mathematical models are just simplifications and approximations to what really happens, so at some stage it becomes pointless to increase the precision of the calculation, because it isn’t increasing the accuracy.

IMO arbitrary cut-offs should be embraced as a recognition that we don’t know what is happening at that level, which would help avoid such nonsense as extrapolating to singularities, then pretending that the singularity is or was a real thing.

You’re right. My thoughts have become much more clear and concise.

The ultraviolet cut-off in quantum mechanics actually cuts off infinity. In the absence of the cut-off, the equations diverge. This is what makes renormalisation necessary.

“The ultraviolet (UV) catastrophe, also called the Rayleigh–Jeans catastrophe, is the prediction of classical electromagnetism that the integrated intensity of the radiation emitted by an ideal black body at thermal equilibrium goes to infinity as wavelength decreases.”

The problem doesn’t just occur in black body radiation but throughout quantum mechanics.

I have now downloaded

https://www.researchgate.net/publication/260773263_Renormalization_Methods_A_Guide_for_Beginners

I may be some time.

The Rev Dodgson said:

I have now downloaded

https://www.researchgate.net/publication/260773263_Renormalization_Methods_A_Guide_for_Beginners

I may be some time.

Quicker than I expected. The download is just the contents pages from a 320 page book.

For nonstandard analysis, two things are urgently needed.

One is an introductory textbook – there isn’t one.

There are a very few books that “claim” to be an introduction to nonstandard analysis, but they need a University maths degree to understand. Whereas some parts of non-standard analysis can be understood in primary school.

The second need is a journal in which papers on nonstandard analysis can be published. When I looked through the literature I found that less than 10% of the important publications are in journals. Most are privately published, monographs. The most recent breakthrough paper could only be published in a “history of philosophy” publication. Yuk. Well-known mathematics and science journals won’t touch the topic.

For Rev D, slides from my youtube on quantum mechanics. Which infinity Part 4

About renormalisation.