5 thoughts on “A universe without math?”

1. Mathematical posture …

   • Caltech Magazine > “Writing in the Language of Math” by Whitney Clavin (Feature, Summer 2022) – What Does Math Mean? [1]

   If math can be expressed in words, what does this say about the true nature of math? Are mathematical symbols merely human inventions used for expressing abstract ideas, or are they part of a fundamental language of the universe that exists independently of us?

   Christopher Hitchcock, the J. O. and Juliette Koepfli Professor of Philosophy, says there are two main camps when it comes to the meaning of math and numbers. The first group follows ideas put forth by the ancient Greek philosopher Plato, who believed mathematical objects are real and possess identities that exist beyond ourselves.

   Feynman … said in a lecture at Cornell University in 1964: “To those who do not know mathematics, … If you want to learn about nature, to appreciate nature, it is necessary to understand the language she speaks in.”

   But as Hitchcock explains, other philosophers believe math is merely an invented tool of science and not an inherent part of our reality. Hartry Field of New York University has gone so far as to reformulate Newtonian mechanics without references to numbers. Field argues the structures described by math are fictitious and not literally true.

   Notes

   [1] Compare this YouTube video by Sabine Hossenfelder.

   • YouTube > Sabine Hossenfelder > “Are we made of math?” (Jul 31, 2021)

   We use the same notion of “reality” in physics, that something is real because it’s a good explanation for our observations. I am not trying to tell you that this is The Right Way to define reality, it’s just for all I can tell how we use the word.

   … when physicists say that space-time is real or the Higgs-boson is real, they mean that a certain mathematical structure correctly describes observations.

   But just because you have math for something doesn’t mean it’s real. … just that this is a belief-based statement, not supported by evidence.

   

2. Mathematical models gone awry?

   • Wired > “Have Some Scientists Gotten Too Excited About the Multiverse?” by Geek’s Guide To The Galaxy (9-9-2022) – Episode 525 of the Geek’s Guide to the Galaxy podcast, an interview with Sabine Hossenfelder

   Sabine Hossenfelder is a theoretical physicist and creator of the popular YouTube series Science Without the Gobbledygook. In her new book Existential Physics, she argues that some of her colleagues may have gotten a little too excited about wild ideas like multiverse theory or the simulation hypothesis.

   Multiverse theory states that an infinite number of alternate universes are constantly branching off from our own. Hossenfelder says it’s possible to create mathematical models that are consistent with multiverse theory, but that doesn’t necessarily tell you anything about reality. “I know quite a lot of cosmologists and astrophysicists who actually believe that other universes are real, and I think it’s a misunderstanding of how much mathematics can actually do for us,” she says.

   
   Credit: Pixabay/CC0 Public Domain

3. As noted elsewhere, physical models (and associated supercomputer simulations) keep getting bigger and more complicated. There’s an implicit mantra that “more detail is better.”

   This article questions that assumption. Particularly regarding national epidemiological models, global hydrological models.

   The article also reminds me of more and more articles touting the value of AI systems but bereft of an intelligible predictive model.

   • Phys.org > “How a quest for mathematical truth and complex models can lead to useless scientific predictions” by Arnald Puy, The Conversation (November 5, 2022) – If more detailed models produce sharper estimates and better predictions, does that mean that they are closer to reality?

   (quote) A dominant view in science is that there is a mathematical truth structuring the universe. It is assumed that the scientist’s job is to decipher these mathematical relations: once understood, they can be translated into mathematical models [1]. Running the resulting “silicon reality” in a computer may then provide us with useful insights into how the world works.

   Once these new additions [parameters and mechanisms] and their associated uncertainties are integrated into the model, they pile on top of the uncertainties already there. And uncertainties keep on expanding with every model upgrade, making the model output fuzzier at every step of the way—even if the model itself becomes more faithful to reality.

   Notes

   [1] Cf. Wiki > Allegory of the cave – “A philosopher aims to understand and perceive the higher levels of reality.”

   (quote) The shadows are the prisoners’ [our everyday] reality, but are not accurate representations of the real world. The shadows represent the fragment of reality that we can normally perceive through our senses, while the objects under the sun represent the true forms of objects that we can only perceive through reason. Three higher levels exist: the natural sciences; mathematics, geometry, and deductive logic; and the theory of forms.

   
   An illustration of The Allegory of the Cave

4. 

   So, here’s an article by a senior theoretical physicist, commenting on the relationship between (increasingly complex) mathematical models and our understanding of the universe. While powerful, our mathematical theories (in physics) are approximate, evolving.

   • New Scientist > “Why the laws of physics don’t actually exist” by theoretical condensed matter physicist Sankar Das Sarma [1] (9 December 2022) – While amazing that we can make sense of some aspects of the universe through the laws of physics, it is difficult to imagine that a thousand years from now physicists will still use quantum mechanics as the fundamental description of nature.

   First things first. What we often call laws of physics are really just consistent mathematical theories that seem to match some parts of nature. … precise and consistent ways of describing the reality we see. This should be obvious from the fact that these laws are not static; they evolve as our empirical knowledge of the universe improves.

   One possible conclusion from this is that the conventional reductionist approach of particle physics, where natural laws are increasingly focused on smaller and smaller building blocks (like molecules, atoms and particles) and fundamental forces (like gravity and electromagnetism) acting between them, is no longer a fruitful way of looking at the physical world.

   I know from my 40 years of experience in working on real-life physical phenomena that the whole idea of an ultimate law based on an equation using just the building blocks and fundamental forces is unworkable and essentially a fantasy. We never know precisely which equation describes a particular laboratory situation. Instead, we always have to build models and approximations to describe each phenomenon even when we know that the equation controlling it is ultimately some form of the Schrödinger equation!

   … the standard model of particle physics, the theory of superconductivity and the theory of atomic spectra are all built using the rules of quantum mechanics, but they have little to do with each other. In addition, space and time are variables that have to be put in by hand into the theory, when space and time should come out naturally from any ultimate law of physics. This has remained perhaps the greatest mystery in fundamental physics with no solution in sight.

   Terms

   String theory – the landscape problem, the multiverse
   Reductionism

   Notes

   [1] Sankar Das Sarma is a theoretical physicist based at the University of Maryland, College Park. His interests are diverse, spanning the strange properties of matter to how information should be understood in the quantum realm.

   In 2010, Das Sarma and collaborators, made a prediction that Majorana fermions will be found in condensed matter, in particular, in semiconductor nanowires. This has led to considerable experimental activity, led by Microsoft Corporation, to produce a topological quantum computer.

5. Sabine Hossenfelder remarks in her latest book [1] that:

   It strikes me as presumptuous to think that humans have already discovered the language in which nature speaks, basically on the first try and right after we appeared on the surface of the planet.

   While her arguments “implicitly assume that mathematics itself is timeless, that mathematical truth is eternal, and that logic doesn’t change,” additional comforting beliefs are not scientific – unnecessary (superfluous) to explain what we observe. [2]

   … physics isn’t math. Physics is a science and as such has the purpose of describing observations of natural phenomena. Yes, we use mathematics in physics, and plenty of that, as I’m sure you have noticed. But we do this not because we know the world is truly mathematics. It may be mathematics – this possibility is known as Platonism, but Platonism is a philosophical position, not a scientific one. All we can tell from observations is that math is useful to describe the world.

   However, the belief that reality is math is deeply ingrained into the thinking of many physicists … Physicists may not consciously subscribe to the idea that math is real and when asked will deny it, but in practice they do not distinguish the two. This conflation has consequences, for they sometimes erroneously come to think their math reveals more about reality than it possibly can.

   Notes

   [1] Hossenfelder, Sabine (2022-08-08T23:58:59.000). Existential Physics. Penguin Publishing Group. Kindle Edition.

   [2] She also notes that: “you do not need to silence rational thought to make space for hope, belief, and faith.”

   

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