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diff --git a/math-science-and-philosophy.html b/math-science-and-philosophy.html new file mode 100644 index 0000000..b83b450 --- /dev/null +++ b/math-science-and-philosophy.html @@ -0,0 +1,284 @@ +<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN" "http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd"> +<html xmlns="http://www.w3.org/1999/xhtml" lang="en"> + <head> + <title>Math, Science, and Philosophy</title> + <link rel="stylesheet" href="/style.css" /> + <meta name="viewport" content="width=device-width, initial-scale=1.0" /> + <link rel="shortcut icon" href="/favicon.ico" type="image/x-icon" /> + <meta charset="utf-8" /> + </head> + <body> +<h1>Math, Science, and Philosophy</h1> +<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 emperical +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 whther 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 emperical information in a +concise and rationalized manner.</p> +<p>This naturally leads us to a question: How is math used in +experimental/emperical 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/decribe 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> + <div id="footer"> + <hr /> + <p><a href="/">Runxi Yu's Website</a></p> + + </div> + </body> +</html> |