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<h1>Math, Science, and Philosophy</h1>
<p>This document is still in discussion and may be improved over time.</p>
<p>Article ID: 10</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>
<h2 id="invented-or-discovered">Invented or discovered?</h2>
<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">Pincipia 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, 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.</p>
<p>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.</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 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.</p>
<h2 id="applicability-in-science">Applicability in Science</h2>
<p>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.</p>
<p>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?</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"><em>F</em> = <em>m</em><em>a</em></span> and that
<span
class="math inline"><em>a</em> = 10 m/s<sup>2</sup>, <em>m</em> = 1 kg</span>,
we conclude that <span class="math inline"><em>F</em> = 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). 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 <span class="math inline"><em>π</em></span>
while it's actually less than <span
class="math inline"><em>π</em></span> after the 15<sup>th</sup>
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>
<h2 id="random-ideas">Random Ideas</h2>
<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>What does it even mean to predict physical properties and
theories with mathematical logic?</p></li>
<li><p>Under what circumstance shall mathematical logic be ``trusted'' in
physics?</p></li>
<li><p>How is it possible to know <em>anything</em> 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 <em>are</em> true
statements that we cannot prove, and there <em>may be</em>
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"><em>A</em></span> (i. e. physics is squishy) is both
true and false. Thus, <span class="math inline"><em>A</em> = 1</span>
and <span class="math inline"><em>A</em> = 0</span> are both true. Then,
take a random statement <span class="math inline"><em>B</em></span>
(let's say ``my friend likes humanities''). Thus we have <span
class="math inline"><em>A</em> + <em>B</em> = 1</span> where <span
class="math inline">+</span> is a boolean ``or'' operator because <span
class="math inline"><em>A</em> = 1</span> and <span
class="math inline">1 + <em>x</em> = 1</span> (<span
class="math inline"><em>x</em></span> is any statement). But then
because <span class="math inline"><em>A</em> = 0</span>, thus <span
class="math inline">0 + <em>B</em> = 1</span>, which means that <span
class="math inline"><em>B</em></span> must be 1 (if <span
class="math inline"><em>B</em></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 ``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.</p></li>
</ul>
<h2 class="unnumbered" id="bugs">Bugs</h2>
<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>
<h2 class="unnumbered" id="acknowledgements">Acknowledgements</h2>
<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 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.</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 extremely superficially as I have no experience in
particle physics or in special unitary groups.<a href="#fnref1"
class="footnote-back" role="doc-backlink">↩︎</a></p></li>
</ol>
</section>
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