This document is still in discussion and may be improved over time.
Article ID: 10
Utilities developed in mathematics are often used to apply theories of the sciences, such as the use of basic arithmetic, calculus, complex analysis, and everything in between in empirical/experimental sciences such as physics. We often take for granted that mathematics as we know it today would work in the sciences. However, considering my impression of math as formally being a creation and natural sciences being mostly observant, it is worth questioning the linkage between these subjects, and whether our use of mathematics, especially in the prediction of theories of physics, is logically linked to the physics itself, or just so happens to a coincidence which we ought to explain.
This article attempts to address these questions, but cannot provide a full answer, for which extensive research would be required which time does not allow for. Rather, this shall be treated as a brief brain-teaser, which discussions may evolve from the text itself, or from the various editorial footnotes and bugs. I would like to, afterwards, complete this article and make it comprehensive and structured, but I'll need ideas from the discussion.
Initially, it feels like mathematics is a pure invention of the human mind. Formal definitions of mathematical systems (albeit unsuccessful in creating the complete and consistent system intended) such as that presented in Pincipia Mathematica do not refer to any tangible objects and are purely conceptual. Deriving theorems from axioms and other theorems, applying general theorems to specific conditions, etc. are all, formally, abstract activities with little reference to the physical world.
However, humans do not truly invent ideas out of pure thought. The basic building blocks of our analytical cognition, which may be in some sense considered ``axioms'' of our perspective of the world, results from us observing the world around us, finding patterns, which then evolve into abstract ideas. Consider the possibility that the formation of numbers as a concept in mathematics results from humans using primitive ideas that resemble numbers to count and record enumerations of discrete objects. Then as people had the need to express non-integer amounts, fractions, and later decimals (or primitive ideas and representations thereof), were born. Previously discrete concepts, numbers, are now used to represent values on continuous spectrums, such as volume, mass, etc. But then consider an alternative world where we are jellyfish swimming through blank water: although this concept of volume is applicable to blank water, it is arguable whether the numeric representation and thus the concept of numerical volume would exist in the first place with the absence of discrete objects. This is an example on how human sense perception affects the process for which we invent mathematics, even if the formal definition thereof does not refer to tangible objects, not to mention how many mathematical constructs such as calculus were specifically created to solve physics problems but is defined in terms of pure math.
Ultimately, even formally defined axiomatic systems have their axioms based on human intuition, which in turn is a result of perspective observing of the natural world.
Additionally, let's take the time to appreciate how well often mathematical concepts, formally defined by human intuition and logic, map to experimentally verifiable physical concepts. This further suggests how natural sciences has an effect on mathematics. (See Section [applicability-in-science] for details.)
The way I like to think about it is: The system of mathematics is formally an invention, but the intuition that led to the axioms, and what theorems we think about and prove, are the result of human discovery. There are both elements to it, and a dichotomous classification would be inappropriate.
Despite how mathematics was likely inspired by tangible perception, the vast majority of modern formal mathematical constructs originate theoretically. In fact, as seen with the use of complex Hilbert space in quantum mechanics, mathematical concepts are sometimes developed much earlier than a corresponding physics theory which utilizes it extensively. It is impressive how formal creations of humans' intuition for beauty in pure math has such a mapping and reflection in the real world.
This naturally leads us to a question: How is math used in experimental sciences? Why? Is that use consistent and based logically, or would it possibly be buggy?
I believe that mathematics has two main roles in physics. The first is calculations, often as an abstraction of experimental experience into a general formula, which is then applied to specific questions. With the knowledge that F = ma and that a = 10 m/s2, m = 1 kg, we conclude that F = 10 N. But many times this involves or implies the second role of math in physics, because calculations depend on corresponding concepts, and sometimes the mathematical utilities themselves are developed from physics but are defined in terms of pure math (such as calculus): physicists analogize mathematical concepts with tangible physical objects and physics concepts, and think about the physical world in a mathematically abstract way. For example, the SU(3) group which finds it origins in the beauties of pure math (group theory is inherently about symmetry), is used extensively in the physics of elementary particles to represent particle spin.1 But for the latter of these use-cases, I am skeptical. Mathematics as we know it is incomplete (Gödel's first incompleteness theorem, in summary, proves that any system of mathematics with Peano Arithmetic cannot prove all true statements in its own system), possibly inconsistent (Gödel's second incompleteness theorem, in summary, proves that any system of mathematics with Peano Arithmetic cannot prove its own consistency), and is somewhat unpredictable (Turing's halting problem, basically saying that it is impossible to, without running the algorithm itself, predict whether a general algorithm would halt or would run forever). We haven't found major loopholes for inconsistency yet, but it is astonishing how mathematics, a system of such theoretical imperfection, is used in every part of physics, not just for its calculations but also for representation of ideas down to the basic level. I find this to be uncanny. What if the physics theories we derive are erroneous because of erroneous mathematical systems or concepts? I believe that part of the answer is ``experiments'', to return to the empirical nature of, well, empirical sciences, and see if the theories actually predict the results. But there are tons of logistical issues that prevent us from doing so, not to mention the inherent downside to experiments: a limited number of attempts cannot derive a general-case theory (take the Borwein integral as an example: a limited number of experiments may easily conclude that it's always π while it's actually less than π after the 15th iteration). So then, we turn to logical proof. But then because mathematical logic is incomplete, we are not guaranteed to be able to prove a given conjecture, which may be otherwise indicated by experiments, to be correct.
Here are some of my random ideas that I haven't sorted into fully-explained paragraphs due to the lack of time to do so. However, I believe that the general point is here, and I would appreciate a discussion about these topics.
What does it even mean to predict physical properties and theories with mathematical logic?
Under what circumstance shall mathematical logic be ``trusted'' in physics?
How is it possible to know anything in physics? Experiments can be inaccurate or conducted wrongly or can be affected by physical properties completely unknown to us, and mathematical proof can be erroneous because of systematic flaws and/or false assumptions about the representation of physical entities in math.
Gödel's theorems only tell us that there are true statements that we cannot prove, and there may be inconsistencies. My intuition suggests that these statements and inconsistencies would be in the highly theoretical realm of math, which if accurately identified and are avoided in physics, would not pose a threat to applied mathematics in physics.
However, it shall be noted that any single inconsistency may be abused to prove any statement, if consistencies were to be found in math: Suppose that we know a statement A (i. e. physics is squishy) is both true and false. Thus, A = 1 and A = 0 are both true. Then, take a random statement B (let's say ``my friend likes humanities''). Thus we have A + B = 1 where + is a boolean ``or'' operator because A = 1 and 1 + x = 1 (x is any statement). But then because A = 0, thus 0 + B = 1, which means that B must be 1 (if B is zero, then 0 + 0 = 0). Thus, if we can prove that ``physics is squishy'' and ``physics is not squishy'' (without differences in definition), then we can literally prove that ``my friend likes humanities''. Other from not defining subjective things like ``squishy'' and ``is'' (in terms of psychology), we can't get around this easily, and everything would be provable, which would not be fun for physics.
No citations present for referenced materials. Thus, this article is not fit for publication, and shall not be considered an authoritative resource. The addition of references will massively improve the status of this article.
The ideas are a bit messy. The structure needs to be reorganized. Repetition is prevalent and must be reduced to a minimum.
Multiple documents were consulted in the writing of this article, which sometimes simply summarizes ideas already expressed by others. Please see the attached reading materials for details. Works of Eugene Wigner were especially helpful.
Contributors include many YK Pao School students and faculty. Insightful conversations with friends have given me great inspiration in the ideas expressed in this article and discussions are still ongoing. For privacy reasons their names aren't listed, but I would be happy to put names on here at request/suggestion.
I'm not exactly sure about this, though, I can only comprehend it extremely superficially as I have no experience in particle physics or in special unitary groups.↩︎