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diff --git a/article/math-science-and-philosophy.html b/article/math-science-and-philosophy.html deleted file mode 100644 index 94f0035..0000000 --- a/article/math-science-and-philosophy.html +++ /dev/null @@ -1,284 +0,0 @@ -<!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>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> -<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> |