I totally agree. I totally agree. I hope saying it twice will convince you that I mean it.
When infinities arise in physics equations, it doesn’t mean there’s a physical infinity. It means that our physics has broken down. Our equations don’t apply. I totally get that. In fact even our friend Max gets that.
blogs.discovermagazine.com/crux/ … g-physics/
The point I am making is something different. I am pointing out that:
All of our modern theories of physics rely ultimately on highly abstract infinitary mathematics
That doesn’t mean that they necessarily do; only that so far, that’s how the history has worked out. There is at the moment no credible alternative. There are attempts to build physics on constructive foundations (there are infinite objects but they can be constructed by algorithms). But not finitary principles, because to do physics you need the real numbers; and to construct the real numbers we need infinite sets.
I collected some examples of the infinitary math underlying physics. I tried to be brief. Each example could be expanded to a book or the work of a lifetime. I’ll do my best to answer specific questions. As with Fubini I regret that it’s beyond me to explain any of these examples fully and in detail with perfect clarity and without requiring effort on the part of the reader. That’s what TED talks are for. /s
- The rigorization of Newton’s calculus culminated with infinitary set theory.
Newton discovered his theory of gravity using calculus, which he invented for that purpose. However, it’s well-known that Newton’s formulation of calculus made no logical sense at all. If (\Delta y) and (\Delta x) are nonzero, then (\frac{\Delta y}{\Delta x}) isn’t the derivative. And if they’re both zero, then the expression makes no mathematical sense! But if we pretend that it does, then we can write down a simple law that explains apples falling to earth and the planets endlessly falling around the sun.
It took another 200 years for mathematicians to develop a rigorous account of calculus from first principles; and those first principles are infinitary set theory. No set theory, no real numbers, no calculus, no gravity.
encyclopediaofmath.org/inde … f_analysis
- Einstein’s gneral relativity uses Riemann’s differential geometry.
In the 1840’s Bernhard Riemann developed a general theory of surfaces that could be Euclidean or very far from Euclidean. As long as they were “locally” Euclidean. Like spheres, and torii, and far weirder non-visualizable shapes. Riemann showed how to do calculus on those surfaces. 60 years later, Einstein had these crazy ideas about the nature of the universe, and the mathematician Minkowski saw that Einstein’s ideas made the most mathematical sense in Riemann’s framework. This is all abstract infinitary mathematics.
en.wikipedia.org/wiki/Differential_geometry
en.wikipedia.org/wiki/Introduct … relativity
- Fourier series link the physics of heat to the physics of the Internet; via infinite trigonometric series.
In 1807 Joseph Fourier analyzed the mathematics of the distribution of heat through an iron bar. He discovered that any continuous function can be expressed as an infinite trigonometric series, which looks like this:
$$f(x) = \sum_{n=0}^\infty a_n \cos(nx) + \sum_{n=1}^\infty b_n \sin(nx)$$
I only posted that because if you managed to survive high school trigonometry, it’s not that hard to unpack. You’re composing any motion into a sum of periodic sine and cosine waves, one wave for each whole number frequency. And this is an infinite series of real numbers, which we cannot make sense of without using infinitary math.
Fast forward to present time. Fourier series underlie the propagation of digital signals over the Internet. They allow us to converse in this very moment.
en.wikipedia.org/wiki/Fourier_series
- Quantum theory is functional analysis.
If you took linear algebra, then functional analysis can be thought of as infinite-dimensional linear algebra combined with calculus. Functional analysis studies spaces whose points are actually functions; so you can apply geometric ideas like length and angle to wild collections of functions. In that sense functional analysis actually generalizes Fourier series.
Quantum mechanics is expressed in the mathematical framework of functional analysis. QM takes place in an infinite-dimensional Hilbert space. To explain Hilbert space requires a deep dive into modern infinitary math. In particular, Hilbert space is complete, meaning that it has no holes in it. It’s like the real numbers and not like the rational numbers.
QM rests on the mathematics of uncountable sets, in an essential way.
ps – There’s our buddy Hilbert again. He did many great things. William Lane Craig misuses and abuses Hilbert’s popularized example of the infinite hotel to make disingenuous points about theology and in particular to argue for the existence of God. That’s what I’ve got against Craig.
- Cantor was led to set theory from Fourier series.
In every online overview of Georg Cantor’s magnificent creation of set theory, nobody ever mentions how he came upon his ideas. It’s as if he woke up one day and decided to revolutionize the foundations of math and piss off his teacher and mentor Kronecker. Nothing could be further from the truth.
Cantor was in fact studing Fourier’s trigonometric series! One of the questions of that era was whether a given function could have more than one distinct Fourier series. To investigate this problem, Cantor had to consider the various types of sets of points on which two series could agree; or equivalently, the various sets of points on which a trigonometric series could be zero. He was thereby led to the problem of classifying various infinite sets of real numbers; and that led him to the discovery of transfinite ordinal and cardinal numbers. (Ordinals are about order in the same way that cardinals are about quantity).
In other words, and this is a fact that you probably will not find stated as clearly as I’m stating it here:
If you begin by studying the flow of heat through an iron rod; you will inexorably discover transfinite set theory.
I do not know what that means in the ultimate scheme of things. But I submit that even the most ardent finitist must at least give consideration to this historical reality.
ias.ac.in/article/fulltext/ … /0977-0999
Conclusion
I hope I’ve been able to explain why I completely agree with your point that infinities in physical equations don’t imply the actual existence of infinities. Yet at the same time, I am pointing out that our best THEORIES of physics are invariably founded on highly infinitary math. As to what that means … for my own part, I can’t help but feel that mathematical infinity is telling us something about the world. We just don’t know yet what that is.