Math, Science, and Philosophy

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Math, Science, and Philosophy (9)
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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.

Invented or discovered?

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.

Applicability in Science

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 15^th +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.

Random Ideas

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 ``Joey 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 ``Joey 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.

Bugs

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.

Acknowledgements

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.↩︎