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<h1>Math, Science, and Philosophy</h1>
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<article>
<p>This document is still in discussion and may be improved over time.</p>
<p>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.</p>
<p>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.</p>
<h1 id="invented-or-discovered">Invented or discovered?<span
id="invented-or-discovered" label="invented-or-discovered"></span></h1>
<p>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 <span class="smallcaps">Principia Mathematica</span> 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.</p>
<p>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, result 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,
concepts such as fractions and decimals (or primitive ideas and
representations thereof), were born. Previously <i>discrete</i>
concepts, numbers, are now used to represent values on
<i>continuous</i> 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 concepts in
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.</p>
<p>Ultimately, even formally defined axiomatic systems have their axioms
based on human intuition, which in turn is a result of empirical
perspective observing of the natural, physical world.</p>
<p>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 <a href="#applicability-in-science" data-reference-type="ref"
data-reference="applicability-in-science">[applicability-in-science]</a>
for details.)</p>
<p>The way I like to think about whether math is an invention or a
discovery 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.</p>
<h1 id="applicability-in-science">Applicability in Science<span
id="applicability-in-science"
label="applicability-in-science"></span></h1>
<p>Despite how mathematics was likely inspired by tangible perception,
the vast majority of modern formal mathematical constructs are defined
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 and how physics tends to formalize empirical information in a
concise and rationalized manner.</p>
<p>This naturally leads us to a question: How is math used in
experimental/empirical sciences? Why? Is that use consistent and based
logically, or would it possibly be buggy?</p>
<p>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 <span
class="math inline"><i>F</i> = <i>m</i><i>a</i></span> and that
<span
class="math inline"><i>a</i> = 10 m/s<sup>2</sup>, <i>m</i> = 1 kg</span>,
we conclude that <span class="math inline"><i>F</i> = 10 N</span>. 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 <span class="math inline">SU(3)</span>
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.<a href="#fn1"
class="footnote-ref" id="fnref1" role="doc-noteref"><sup>1</sup></a> 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, and thus
there is no general algorithm to predict whether an algorithm will halt
in finite time). We haven’t found major loopholes for inconsistency yet,
but it is astonishing howmathematics, 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 <span
class="math inline"><i>π</i></span> while it’s actually less than
<span class="math inline"><i>π</i></span> 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.</p>
<p>Note that I am not arguing that physics derives its concepts from
mathematics; I believe that physics has chosen the part of math that it
believes to be helpful for use therein. However, these have strange and
unforseeable implications.</p>
<p>The addition of mathematical concepts into physics doesn’t only bring
the maths we want to bring over, it brings all relevant definitions,
axioms, logic, proofs, theorems, etc. all along with it. Once we
"assign" that a physical entity is "represented" by a "corresponding
concept" in mathematics, we can only abide by the development thereof.
So although physics originally isn’t guided by mathematics, the act of
choosing the part of math that’s useful in physics puts physics under
the iron grip of mathematical logic, which is inconsistent and
potentially incomplete, as contrary to the realistic and observable
nature that physics is supposed to be.</p>
<p>I had a brief chat with Mr. Coxon and he aclled how the existence of
neutrinos were predicted "mathematically" before they were
experimentally discovered physically. I do not know the history of all
this, but Mr. Coxon said that physicists
looked at a phenomenon (I believe that was beta decay) and went like:
"where did that missing energy go"? and proposed that there was a
particle called a neutrino that fills in the missing gap.
(Alternatively, they could have challenged the conservation of energy,
which leads us to the topic of "why do we find it so hard to challenge
theories that seem beautiful, and why does conservation and symmetry
seem beautiful", but let’s get back on topic...) Then twnety years later
neutrinos were "discovered" physically by experiments. Mr. Coxon said
that it looked like that mathematics predicted and in some resepct
"guided" physics. Personally I believe that this isn’t a purely
"mathematical" pre-discovery and it’s more of a "conservation of energy,
a physics theory was applied, and math was used as a utility to find
incompletenesses in our understanding of particles." I think that I’ve
heard (but cannot recall at the moment) two cases where conceptual
analysis in "pure math" perfectly corresponds to the phenomenon in
physics discovered later which again makes me question whether math
played some role in the experiment-phenomenon-discovery cycle of
physics. I guess I need more examples.</p>
<p>I remember that Kant argued that human knowledge is human perception
and its leading into rational thought and reason. To me this sounds like
the development of math, but in some sense this could also apply to
physics, though I still believe that physics theories even if reasoned
require experimental "testing" (not "verification") for it to be
acceptable in terms of physics. THis leaves me in a situation where none
of the ways of knowing that I can understand, even if used together,
could bring about an absolutely correct[tm] theory of physics. See,
reason is flawed because logic may fail, not to mention when we are
literally trying to define/describe novel physics concepts/entities and
there aren’t any definitions to begin with to even start with reasoning
and all we could do is using intuition in discovery. (Pattern finding in
intuitive concepts would require formalization to be somewhat
acceptable, but not absolutely ground-standing, in the realm of reason.)
And then, experiments are flawed because errors will always exist in the
messey real world (and if we do simulations that’s just falling back to
our existing understanding of logical analysis). So now we have no
single way, or combination of methods, to accurately verify the
correctness of a physics theory, which by definition of physical is
representative of the real world, basically saying that "we will never
know how things work in the real world". That feels uncanny. Also, how
do I even make sense of a physics theory to be "correct"? It’s arguable
whether any physics theory could be correct in the first place. If Kant
is correct then all our theories of physics is ultimately perception and
having biology in the form of human observations in the absolute and
hard-core feeling of physics is so weird.</p>
<h1 id="random-ideas">Random Ideas</h1>
<p>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.</p>
<ul>
<li><p>How is it possible to know <i>anything</i> 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.</p></li>
<li><p>Gödel’s theorems only tell us that there <i>are</i> true
statements that we cannot prove, and there <i>may be</i>
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.</p>
<p>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 <span
class="math inline"><i>A</i></span> (i. e. physics is squishy) is both
true and false. Thus, <span class="math inline"><i>A</i> = 1</span>
and <span class="math inline"><i>A</i> = 0</span> are both true. Then,
take a random statement <span class="math inline"><i>B</i></span>
(let’s say "Z likes humanities"). Thus we have <span
class="math inline"><i>A</i> + <i>B</i> = 1</span> where <span
class="math inline">+</span> is a boolean "or" operator because <span
class="math inline"><i>A</i> = 1</span> and <span
class="math inline">1 + <i>x</i> = 1</span> (<span
class="math inline"><i>x</i></span> is any statement). But then
because <span class="math inline"><i>A</i> = 0</span>, thus <span
class="math inline">0 + <i>B</i> = 1</span>, which means that <span
class="math inline"><i>B</i></span> must be 1 (if <span
class="math inline"><i>B</i></span> is zero, then <span
class="math inline">0 + 0 = 0</span>). Thus, if we can prove that
"physics is squishy" and "physics is not squishy" (without differences
in definition), then we can literally prove that "Z 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.</p></li>
</ul>
<h1 class="unnumbered" id="bugs">Bugs</h1>
<ul>
<li><p>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.</p></li>
<li><p>The ideas are a bit messy. The structure needs to be reorganized.
Repetition is prevalent and must be reduced to a minimum.</p></li>
</ul>
<h1 class="unnumbered" id="acknowledgements">Acknowledgements</h1>
<p>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.</p>
<p>Contributors include many YK Pao School students and faculty.
Insightful conversations with other students, such as MuonNeutrino_,
have given me great inspiration in the ideas
expressed in this article and discussions are still ongoing. For privacy
reasons other names aren’t listed, but I would be happy to put names on
here at request/suggestion.</p>
<section id="footnotes" class="footnotes footnotes-end-of-document"
role="doc-endnotes">
<hr />
<ol>
<li id="fn1"><p>I’m not exactly sure about this, though, I can only
comprehend it slightly superficially as I don't have much experience in
particle physics or in special unitary groups, yet.
<a
href="#fnref1" class="footnote-back" role="doc-backlink">↩︎</a></p></li>
</ol>
</section>
</article>
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