dimitri/blog
1
0
Fork 0
Code Issues Pull requests Projects Releases Packages Wiki Activity Actions

Update references in Peano post

Browse source
This commit is contained in:
Dimitri Lozeve 2019-04-03 21:38:33 +02:00
parent beda8aa4d8
commit 275e10e3df
4 changed files with 65 additions and 54 deletions
Download patch file Download diff file Expand all files Collapse all files

26
_site/atom.xml
View file

@ -65,7 +65,7 @@
</section> </section>
<section> <section>
<h1 id="introduction">Introduction</h1> <h1 id="introduction">Introduction</h1>
<p>I have recently bought the book <em>Category Theory</em> from Steve Awodey <a href="#ref-1">(1)</a> (which is awesome, but probably the topic for another post), and a particular passage excited my curiosity:</p> <p>I have recently bought the book <em>Category Theory</em> from Steve Awodey <span class="citation" data-cites="awodeyCategoryTheory2010">(Awodey 2010)</span> is awesome, but probably the topic for another post), and a particular passage excited my curiosity:</p>
<blockquote> <blockquote>
<p>Let us begin by distinguishing between the following things: i. categorical foundations for mathematics, ii. mathematical foundations for category theory.</p> <p>Let us begin by distinguishing between the following things: i. categorical foundations for mathematics, ii. mathematical foundations for category theory.</p>
<p>As for the first point, one sometimes hears it said that category theory can be used to provide “foundations for mathematics,” as an alternative to set theory. That is in fact the case, but it is not what we are doing here. In set theory, one often begins with existential axioms such as “there is an infinite set” and derives further sets by axioms like “every set has a powerset,” thus building up a universe of mathematical objects (namely sets), which in principle suffice for “all of mathematics.”</p> <p>As for the first point, one sometimes hears it said that category theory can be used to provide “foundations for mathematics,” as an alternative to set theory. That is in fact the case, but it is not what we are doing here. In set theory, one often begins with existential axioms such as “there is an infinite set” and derives further sets by axioms like “every set has a powerset,” thus building up a universe of mathematical objects (namely sets), which in principle suffice for “all of mathematics.”</p>
@ -73,7 +73,7 @@
<p>This statement is interesting because one often considers category theory as pretty “fundamental”, in the sense that it has no issue with considering what I call “dangerous” notions, such as the category <span class="math inline">\(\mathbf{Set}\)</span> of all sets, and even the category <span class="math inline">\(\mathbf{Cat}\)</span> of all categories. Surely a theory this general, that can afford to study such objects, should provide suitable foundations for mathematics? Awodey addresses these issues very explicitly in the section following the quote above, and finds a good way of avoiding circular definitions.</p> <p>This statement is interesting because one often considers category theory as pretty “fundamental”, in the sense that it has no issue with considering what I call “dangerous” notions, such as the category <span class="math inline">\(\mathbf{Set}\)</span> of all sets, and even the category <span class="math inline">\(\mathbf{Cat}\)</span> of all categories. Surely a theory this general, that can afford to study such objects, should provide suitable foundations for mathematics? Awodey addresses these issues very explicitly in the section following the quote above, and finds a good way of avoiding circular definitions.</p>
<p>Now, I remember some basics from my undergrad studies about foundations of mathematics. I was told that if you could define arithmetic, you basically had everything else “for free” (as Kronecker famously said, “natural numbers were created by God, everything else is the work of men”). I was also told that two sets of axioms existed, the <a href="https://en.wikipedia.org/wiki/Peano_axioms">Peano axioms</a> and the <a href="https://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_theory">Zermelo-Fraenkel</a> axioms. Also, I should steer clear of the axiom of choice if I could, because one can do <a href="https://en.wikipedia.org/wiki/Banach%E2%80%93Tarski_paradox">strange things</a> with it, and it is equivalent to many <a href="https://en.wikipedia.org/wiki/Zorn%27s_lemma">different statements</a>. Finally (and this I knew mainly from <em>Logicomix</em>, I must admit), it is <a href="https://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_theorems">impossible</a> for a set of axioms to be both complete and consistent.</p> <p>Now, I remember some basics from my undergrad studies about foundations of mathematics. I was told that if you could define arithmetic, you basically had everything else “for free” (as Kronecker famously said, “natural numbers were created by God, everything else is the work of men”). I was also told that two sets of axioms existed, the <a href="https://en.wikipedia.org/wiki/Peano_axioms">Peano axioms</a> and the <a href="https://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_theory">Zermelo-Fraenkel</a> axioms. Also, I should steer clear of the axiom of choice if I could, because one can do <a href="https://en.wikipedia.org/wiki/Banach%E2%80%93Tarski_paradox">strange things</a> with it, and it is equivalent to many <a href="https://en.wikipedia.org/wiki/Zorn%27s_lemma">different statements</a>. Finally (and this I knew mainly from <em>Logicomix</em>, I must admit), it is <a href="https://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_theorems">impossible</a> for a set of axioms to be both complete and consistent.</p>
<p>Given all this, I realised that my knowledge of foundational mathematics was pretty deficient. I do not believe that it is a very important topic that everyone should know about, even though Gödel’s incompleteness theorem is very interesting from a logical and philosophical standpoint. However, I wanted to go deeper on this subject.</p> <p>Given all this, I realised that my knowledge of foundational mathematics was pretty deficient. I do not believe that it is a very important topic that everyone should know about, even though Gödel’s incompleteness theorem is very interesting from a logical and philosophical standpoint. However, I wanted to go deeper on this subject.</p>
<p>In this post, I will try to share my path through Peano’s axioms <a href="#ref-2">(2)</a>, because they are very simple, and it is easy to uncover basic algebraic structure from them.</p> <p>In this post, I will try to share my path through Peano’s axioms <span class="citation" data-cites="gowersPrincetonCompanionMathematics2010">(Gowers, Barrow-Green, and Leader 2010)</span>, because they are very simple, and it is easy to uncover basic algebraic structure from them.</p>
<h1 id="the-axioms">The Axioms</h1> <h1 id="the-axioms">The Axioms</h1>
<p>The purpose of the axioms is to define a collection of objects that we will call the <em>natural numbers</em>. Here, we place ourselves in the context of <a href="https://en.wikipedia.org/wiki/First-order_logic">first-order logic</a>. Logic is not the main topic here, so I will just assume that I have access to some quantifiers, to some predicates, to some variables, and, most importantly, to a relation <span class="math inline">\(=\)</span> which is reflexive, symmetric, transitive, and closed over the natural numbers.</p> <p>The purpose of the axioms is to define a collection of objects that we will call the <em>natural numbers</em>. Here, we place ourselves in the context of <a href="https://en.wikipedia.org/wiki/First-order_logic">first-order logic</a>. Logic is not the main topic here, so I will just assume that I have access to some quantifiers, to some predicates, to some variables, and, most importantly, to a relation <span class="math inline">\(=\)</span> which is reflexive, symmetric, transitive, and closed over the natural numbers.</p>
<p>Without further digressions, let us define two symbols <span class="math inline">\(0\)</span> and <span class="math inline">\(s\)</span> (called <em>successor</em>) such that:</p> <p>Without further digressions, let us define two symbols <span class="math inline">\(0\)</span> and <span class="math inline">\(s\)</span> (called <em>successor</em>) such that:</p>
@ -157,16 +157,22 @@ then <span class="math inline">\(\varphi(n)\)</span> is true for every natural n
<h1 id="going-further">Going further</h1> <h1 id="going-further">Going further</h1>
<p>We have imbued our newly created set of natural numbers with a significant algebraic structure. From there, similar arguments will create more structure, notably by introducing another operation <span class="math inline">\(\times\)</span>, and an order <span class="math inline">\(\leq\)</span>.</p> <p>We have imbued our newly created set of natural numbers with a significant algebraic structure. From there, similar arguments will create more structure, notably by introducing another operation <span class="math inline">\(\times\)</span>, and an order <span class="math inline">\(\leq\)</span>.</p>
<p>It is now a matter of conventional mathematics to construct the integers <span class="math inline">\(\mathbb{Z}\)</span> and the rationals <span class="math inline">\(\mathbb{Q}\)</span> (using equivalence classes), and eventually the real numbers <span class="math inline">\(\mathbb{R}\)</span>.</p> <p>It is now a matter of conventional mathematics to construct the integers <span class="math inline">\(\mathbb{Z}\)</span> and the rationals <span class="math inline">\(\mathbb{Q}\)</span> (using equivalence classes), and eventually the real numbers <span class="math inline">\(\mathbb{R}\)</span>.</p>
<p>It is remarkable how very few (and very simple, as far as you would consider the induction axiom “simple”) axioms are enough to build an entire theory of mathematics. This sort of things makes me agree with Eugene Wigner <a href="#ref-3">(3)</a> when he says that “mathematics is the science of skillful operations with concepts and rules invented just for this purpose”. We drew some arbitrary rules out of thin air, and derived countless properties and theorems from them, basically for our own enjoyment. (As Wigner would say, it is <em>incredible</em> that any of these fanciful inventions coming out of nowhere turned out to be even remotely useful.) Mathematics is done mainly for the mathematician’s own pleasure!</p> <p>It is remarkable how very few (and very simple, as far as you would consider the induction axiom “simple”) axioms are enough to build an entire theory of mathematics. This sort of things makes me agree with Eugene Wigner <span class="citation" data-cites="wignerUnreasonableEffectivenessMathematics1990">(Wigner 1990)</span> when he says that “mathematics is the science of skillful operations with concepts and rules invented just for this purpose”. We drew some arbitrary rules out of thin air, and derived countless properties and theorems from them, basically for our own enjoyment. (As Wigner would say, it is <em>incredible</em> that any of these fanciful inventions coming out of nowhere turned out to be even remotely useful.) Mathematics is done mainly for the mathematician’s own pleasure!</p>
<blockquote> <blockquote>
<p>Mathematics cannot be defined without acknowledging its most obvious feature: namely, that it is interesting — M. Polanyi <a href="#ref-3">(3)</a></p> <p>Mathematics cannot be defined without acknowledging its most obvious feature: namely, that it is interesting — M. Polanyi <span class="citation" data-cites="wignerUnreasonableEffectivenessMathematics1990">(Wigner 1990)</span></p>
</blockquote> </blockquote>
<h1 id="references">References</h1> <h1 id="references" class="unnumbered">References</h1>
<ol> <div id="refs" class="references">
<li><span id="ref-1"></span>Awodey, Steve. Category Theory. 2nd ed. Oxford Logic Guides 52. Oxford ; New York: Oxford University Press, 2010.</li> <div id="ref-awodeyCategoryTheory2010">
<li><span id="ref-2"></span>Gowers, Timothy, June Barrow-Green, and Imre Leader. The Princeton Companion to Mathematics. Princeton University Press, 2010.</li> <p>Awodey, Steve. 2010. <em>Category Theory</em>. 2nd ed. Oxford Logic Guides 52. Oxford ; New York: Oxford University Press.</p>
<li><span id="ref-3"></span>Wigner, Eugene P. ‘The Unreasonable Effectiveness of Mathematics in the Natural Sciences’. In Mathematics and Science, by Ronald E Mickens, 291–306. World Scientific, 1990. <a href="https://doi.org/10.1142/9789814503488_0018" class="uri">https://doi.org/10.1142/9789814503488_0018</a>.</li> </div>
</ol> <div id="ref-gowersPrincetonCompanionMathematics2010">
<p>Gowers, Timothy, June Barrow-Green, and Imre Leader. 2010. <em>The Princeton Companion to Mathematics</em>. Princeton University Press.</p>
</div>
<div id="ref-wignerUnreasonableEffectivenessMathematics1990">
<p>Wigner, Eugene P. 1990. “The Unreasonable Effectiveness of Mathematics in the Natural Sciences.” In <em>Mathematics and Science</em>, by Ronald E Mickens, 291–306. WORLD SCIENTIFIC. <a href="https://doi.org/10.1142/9789814503488_0018" class="uri">https://doi.org/10.1142/9789814503488_0018</a>.</p>
</div>
</div>
</section> </section>
</article> </article>
]]></summary> ]]></summary>

26
_site/posts/peano.html
View file

@ -31,7 +31,7 @@
</section> </section>
<section> <section>
<h1 id="introduction">Introduction</h1> <h1 id="introduction">Introduction</h1>
<p>I have recently bought the book <em>Category Theory</em> from Steve Awodey <a href="#ref-1">(1)</a> (which is awesome, but probably the topic for another post), and a particular passage excited my curiosity:</p> <p>I have recently bought the book <em>Category Theory</em> from Steve Awodey <span class="citation" data-cites="awodeyCategoryTheory2010">(Awodey 2010)</span> is awesome, but probably the topic for another post), and a particular passage excited my curiosity:</p>
<blockquote> <blockquote>
<p>Let us begin by distinguishing between the following things: i. categorical foundations for mathematics, ii. mathematical foundations for category theory.</p> <p>Let us begin by distinguishing between the following things: i. categorical foundations for mathematics, ii. mathematical foundations for category theory.</p>
<p>As for the first point, one sometimes hears it said that category theory can be used to provide “foundations for mathematics,” as an alternative to set theory. That is in fact the case, but it is not what we are doing here. In set theory, one often begins with existential axioms such as “there is an infinite set” and derives further sets by axioms like “every set has a powerset,” thus building up a universe of mathematical objects (namely sets), which in principle suffice for “all of mathematics.”</p> <p>As for the first point, one sometimes hears it said that category theory can be used to provide “foundations for mathematics,” as an alternative to set theory. That is in fact the case, but it is not what we are doing here. In set theory, one often begins with existential axioms such as “there is an infinite set” and derives further sets by axioms like “every set has a powerset,” thus building up a universe of mathematical objects (namely sets), which in principle suffice for “all of mathematics.”</p>
@ -39,7 +39,7 @@
<p>This statement is interesting because one often considers category theory as pretty “fundamental”, in the sense that it has no issue with considering what I call “dangerous” notions, such as the category <span class="math inline">\(\mathbf{Set}\)</span> of all sets, and even the category <span class="math inline">\(\mathbf{Cat}\)</span> of all categories. Surely a theory this general, that can afford to study such objects, should provide suitable foundations for mathematics? Awodey addresses these issues very explicitly in the section following the quote above, and finds a good way of avoiding circular definitions.</p> <p>This statement is interesting because one often considers category theory as pretty “fundamental”, in the sense that it has no issue with considering what I call “dangerous” notions, such as the category <span class="math inline">\(\mathbf{Set}\)</span> of all sets, and even the category <span class="math inline">\(\mathbf{Cat}\)</span> of all categories. Surely a theory this general, that can afford to study such objects, should provide suitable foundations for mathematics? Awodey addresses these issues very explicitly in the section following the quote above, and finds a good way of avoiding circular definitions.</p>
<p>Now, I remember some basics from my undergrad studies about foundations of mathematics. I was told that if you could define arithmetic, you basically had everything else “for free” (as Kronecker famously said, “natural numbers were created by God, everything else is the work of men”). I was also told that two sets of axioms existed, the <a href="https://en.wikipedia.org/wiki/Peano_axioms">Peano axioms</a> and the <a href="https://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_theory">Zermelo-Fraenkel</a> axioms. Also, I should steer clear of the axiom of choice if I could, because one can do <a href="https://en.wikipedia.org/wiki/Banach%E2%80%93Tarski_paradox">strange things</a> with it, and it is equivalent to many <a href="https://en.wikipedia.org/wiki/Zorn%27s_lemma">different statements</a>. Finally (and this I knew mainly from <em>Logicomix</em>, I must admit), it is <a href="https://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_theorems">impossible</a> for a set of axioms to be both complete and consistent.</p> <p>Now, I remember some basics from my undergrad studies about foundations of mathematics. I was told that if you could define arithmetic, you basically had everything else “for free” (as Kronecker famously said, “natural numbers were created by God, everything else is the work of men”). I was also told that two sets of axioms existed, the <a href="https://en.wikipedia.org/wiki/Peano_axioms">Peano axioms</a> and the <a href="https://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_theory">Zermelo-Fraenkel</a> axioms. Also, I should steer clear of the axiom of choice if I could, because one can do <a href="https://en.wikipedia.org/wiki/Banach%E2%80%93Tarski_paradox">strange things</a> with it, and it is equivalent to many <a href="https://en.wikipedia.org/wiki/Zorn%27s_lemma">different statements</a>. Finally (and this I knew mainly from <em>Logicomix</em>, I must admit), it is <a href="https://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_theorems">impossible</a> for a set of axioms to be both complete and consistent.</p>
<p>Given all this, I realised that my knowledge of foundational mathematics was pretty deficient. I do not believe that it is a very important topic that everyone should know about, even though Gödel’s incompleteness theorem is very interesting from a logical and philosophical standpoint. However, I wanted to go deeper on this subject.</p> <p>Given all this, I realised that my knowledge of foundational mathematics was pretty deficient. I do not believe that it is a very important topic that everyone should know about, even though Gödel’s incompleteness theorem is very interesting from a logical and philosophical standpoint. However, I wanted to go deeper on this subject.</p>
<p>In this post, I will try to share my path through Peano’s axioms <a href="#ref-2">(2)</a>, because they are very simple, and it is easy to uncover basic algebraic structure from them.</p> <p>In this post, I will try to share my path through Peano’s axioms <span class="citation" data-cites="gowersPrincetonCompanionMathematics2010">(Gowers, Barrow-Green, and Leader 2010)</span>, because they are very simple, and it is easy to uncover basic algebraic structure from them.</p>
<h1 id="the-axioms">The Axioms</h1> <h1 id="the-axioms">The Axioms</h1>
<p>The purpose of the axioms is to define a collection of objects that we will call the <em>natural numbers</em>. Here, we place ourselves in the context of <a href="https://en.wikipedia.org/wiki/First-order_logic">first-order logic</a>. Logic is not the main topic here, so I will just assume that I have access to some quantifiers, to some predicates, to some variables, and, most importantly, to a relation <span class="math inline">\(=\)</span> which is reflexive, symmetric, transitive, and closed over the natural numbers.</p> <p>The purpose of the axioms is to define a collection of objects that we will call the <em>natural numbers</em>. Here, we place ourselves in the context of <a href="https://en.wikipedia.org/wiki/First-order_logic">first-order logic</a>. Logic is not the main topic here, so I will just assume that I have access to some quantifiers, to some predicates, to some variables, and, most importantly, to a relation <span class="math inline">\(=\)</span> which is reflexive, symmetric, transitive, and closed over the natural numbers.</p>
<p>Without further digressions, let us define two symbols <span class="math inline">\(0\)</span> and <span class="math inline">\(s\)</span> (called <em>successor</em>) such that:</p> <p>Without further digressions, let us define two symbols <span class="math inline">\(0\)</span> and <span class="math inline">\(s\)</span> (called <em>successor</em>) such that:</p>
@ -123,16 +123,22 @@ then <span class="math inline">\(\varphi(n)\)</span> is true for every natural n
<h1 id="going-further">Going further</h1> <h1 id="going-further">Going further</h1>
<p>We have imbued our newly created set of natural numbers with a significant algebraic structure. From there, similar arguments will create more structure, notably by introducing another operation <span class="math inline">\(\times\)</span>, and an order <span class="math inline">\(\leq\)</span>.</p> <p>We have imbued our newly created set of natural numbers with a significant algebraic structure. From there, similar arguments will create more structure, notably by introducing another operation <span class="math inline">\(\times\)</span>, and an order <span class="math inline">\(\leq\)</span>.</p>
<p>It is now a matter of conventional mathematics to construct the integers <span class="math inline">\(\mathbb{Z}\)</span> and the rationals <span class="math inline">\(\mathbb{Q}\)</span> (using equivalence classes), and eventually the real numbers <span class="math inline">\(\mathbb{R}\)</span>.</p> <p>It is now a matter of conventional mathematics to construct the integers <span class="math inline">\(\mathbb{Z}\)</span> and the rationals <span class="math inline">\(\mathbb{Q}\)</span> (using equivalence classes), and eventually the real numbers <span class="math inline">\(\mathbb{R}\)</span>.</p>
<p>It is remarkable how very few (and very simple, as far as you would consider the induction axiom “simple”) axioms are enough to build an entire theory of mathematics. This sort of things makes me agree with Eugene Wigner <a href="#ref-3">(3)</a> when he says that “mathematics is the science of skillful operations with concepts and rules invented just for this purpose”. We drew some arbitrary rules out of thin air, and derived countless properties and theorems from them, basically for our own enjoyment. (As Wigner would say, it is <em>incredible</em> that any of these fanciful inventions coming out of nowhere turned out to be even remotely useful.) Mathematics is done mainly for the mathematician’s own pleasure!</p> <p>It is remarkable how very few (and very simple, as far as you would consider the induction axiom “simple”) axioms are enough to build an entire theory of mathematics. This sort of things makes me agree with Eugene Wigner <span class="citation" data-cites="wignerUnreasonableEffectivenessMathematics1990">(Wigner 1990)</span> when he says that “mathematics is the science of skillful operations with concepts and rules invented just for this purpose”. We drew some arbitrary rules out of thin air, and derived countless properties and theorems from them, basically for our own enjoyment. (As Wigner would say, it is <em>incredible</em> that any of these fanciful inventions coming out of nowhere turned out to be even remotely useful.) Mathematics is done mainly for the mathematician’s own pleasure!</p>
<blockquote> <blockquote>
<p>Mathematics cannot be defined without acknowledging its most obvious feature: namely, that it is interesting — M. Polanyi <a href="#ref-3">(3)</a></p> <p>Mathematics cannot be defined without acknowledging its most obvious feature: namely, that it is interesting — M. Polanyi <span class="citation" data-cites="wignerUnreasonableEffectivenessMathematics1990">(Wigner 1990)</span></p>
</blockquote> </blockquote>
<h1 id="references">References</h1> <h1 id="references" class="unnumbered">References</h1>
<ol> <div id="refs" class="references">
<li><span id="ref-1"></span>Awodey, Steve. Category Theory. 2nd ed. Oxford Logic Guides 52. Oxford ; New York: Oxford University Press, 2010.</li> <div id="ref-awodeyCategoryTheory2010">
<li><span id="ref-2"></span>Gowers, Timothy, June Barrow-Green, and Imre Leader. The Princeton Companion to Mathematics. Princeton University Press, 2010.</li> <p>Awodey, Steve. 2010. <em>Category Theory</em>. 2nd ed. Oxford Logic Guides 52. Oxford ; New York: Oxford University Press.</p>
<li><span id="ref-3"></span>Wigner, Eugene P. ‘The Unreasonable Effectiveness of Mathematics in the Natural Sciences’. In Mathematics and Science, by Ronald E Mickens, 291–306. World Scientific, 1990. <a href="https://doi.org/10.1142/9789814503488_0018" class="uri">https://doi.org/10.1142/9789814503488_0018</a>.</li> </div>
</ol> <div id="ref-gowersPrincetonCompanionMathematics2010">
<p>Gowers, Timothy, June Barrow-Green, and Imre Leader. 2010. <em>The Princeton Companion to Mathematics</em>. Princeton University Press.</p>
</div>
<div id="ref-wignerUnreasonableEffectivenessMathematics1990">
<p>Wigner, Eugene P. 1990. “The Unreasonable Effectiveness of Mathematics in the Natural Sciences.” In <em>Mathematics and Science</em>, by Ronald E Mickens, 291–306. WORLD SCIENTIFIC. <a href="https://doi.org/10.1142/9789814503488_0018" class="uri">https://doi.org/10.1142/9789814503488_0018</a>.</p>
</div>
</div>
</section> </section>
</article> </article>

26
_site/rss.xml
View file

@ -61,7 +61,7 @@
</section> </section>
<section> <section>
<h1 id="introduction">Introduction</h1> <h1 id="introduction">Introduction</h1>
<p>I have recently bought the book <em>Category Theory</em> from Steve Awodey <a href="#ref-1">(1)</a> (which is awesome, but probably the topic for another post), and a particular passage excited my curiosity:</p> <p>I have recently bought the book <em>Category Theory</em> from Steve Awodey <span class="citation" data-cites="awodeyCategoryTheory2010">(Awodey 2010)</span> is awesome, but probably the topic for another post), and a particular passage excited my curiosity:</p>
<blockquote> <blockquote>
<p>Let us begin by distinguishing between the following things: i. categorical foundations for mathematics, ii. mathematical foundations for category theory.</p> <p>Let us begin by distinguishing between the following things: i. categorical foundations for mathematics, ii. mathematical foundations for category theory.</p>
<p>As for the first point, one sometimes hears it said that category theory can be used to provide “foundations for mathematics,” as an alternative to set theory. That is in fact the case, but it is not what we are doing here. In set theory, one often begins with existential axioms such as “there is an infinite set” and derives further sets by axioms like “every set has a powerset,” thus building up a universe of mathematical objects (namely sets), which in principle suffice for “all of mathematics.”</p> <p>As for the first point, one sometimes hears it said that category theory can be used to provide “foundations for mathematics,” as an alternative to set theory. That is in fact the case, but it is not what we are doing here. In set theory, one often begins with existential axioms such as “there is an infinite set” and derives further sets by axioms like “every set has a powerset,” thus building up a universe of mathematical objects (namely sets), which in principle suffice for “all of mathematics.”</p>
@ -69,7 +69,7 @@
<p>This statement is interesting because one often considers category theory as pretty “fundamental”, in the sense that it has no issue with considering what I call “dangerous” notions, such as the category <span class="math inline">\(\mathbf{Set}\)</span> of all sets, and even the category <span class="math inline">\(\mathbf{Cat}\)</span> of all categories. Surely a theory this general, that can afford to study such objects, should provide suitable foundations for mathematics? Awodey addresses these issues very explicitly in the section following the quote above, and finds a good way of avoiding circular definitions.</p> <p>This statement is interesting because one often considers category theory as pretty “fundamental”, in the sense that it has no issue with considering what I call “dangerous” notions, such as the category <span class="math inline">\(\mathbf{Set}\)</span> of all sets, and even the category <span class="math inline">\(\mathbf{Cat}\)</span> of all categories. Surely a theory this general, that can afford to study such objects, should provide suitable foundations for mathematics? Awodey addresses these issues very explicitly in the section following the quote above, and finds a good way of avoiding circular definitions.</p>
<p>Now, I remember some basics from my undergrad studies about foundations of mathematics. I was told that if you could define arithmetic, you basically had everything else “for free” (as Kronecker famously said, “natural numbers were created by God, everything else is the work of men”). I was also told that two sets of axioms existed, the <a href="https://en.wikipedia.org/wiki/Peano_axioms">Peano axioms</a> and the <a href="https://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_theory">Zermelo-Fraenkel</a> axioms. Also, I should steer clear of the axiom of choice if I could, because one can do <a href="https://en.wikipedia.org/wiki/Banach%E2%80%93Tarski_paradox">strange things</a> with it, and it is equivalent to many <a href="https://en.wikipedia.org/wiki/Zorn%27s_lemma">different statements</a>. Finally (and this I knew mainly from <em>Logicomix</em>, I must admit), it is <a href="https://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_theorems">impossible</a> for a set of axioms to be both complete and consistent.</p> <p>Now, I remember some basics from my undergrad studies about foundations of mathematics. I was told that if you could define arithmetic, you basically had everything else “for free” (as Kronecker famously said, “natural numbers were created by God, everything else is the work of men”). I was also told that two sets of axioms existed, the <a href="https://en.wikipedia.org/wiki/Peano_axioms">Peano axioms</a> and the <a href="https://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_theory">Zermelo-Fraenkel</a> axioms. Also, I should steer clear of the axiom of choice if I could, because one can do <a href="https://en.wikipedia.org/wiki/Banach%E2%80%93Tarski_paradox">strange things</a> with it, and it is equivalent to many <a href="https://en.wikipedia.org/wiki/Zorn%27s_lemma">different statements</a>. Finally (and this I knew mainly from <em>Logicomix</em>, I must admit), it is <a href="https://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_theorems">impossible</a> for a set of axioms to be both complete and consistent.</p>
<p>Given all this, I realised that my knowledge of foundational mathematics was pretty deficient. I do not believe that it is a very important topic that everyone should know about, even though Gödel’s incompleteness theorem is very interesting from a logical and philosophical standpoint. However, I wanted to go deeper on this subject.</p> <p>Given all this, I realised that my knowledge of foundational mathematics was pretty deficient. I do not believe that it is a very important topic that everyone should know about, even though Gödel’s incompleteness theorem is very interesting from a logical and philosophical standpoint. However, I wanted to go deeper on this subject.</p>
<p>In this post, I will try to share my path through Peano’s axioms <a href="#ref-2">(2)</a>, because they are very simple, and it is easy to uncover basic algebraic structure from them.</p> <p>In this post, I will try to share my path through Peano’s axioms <span class="citation" data-cites="gowersPrincetonCompanionMathematics2010">(Gowers, Barrow-Green, and Leader 2010)</span>, because they are very simple, and it is easy to uncover basic algebraic structure from them.</p>
<h1 id="the-axioms">The Axioms</h1> <h1 id="the-axioms">The Axioms</h1>
<p>The purpose of the axioms is to define a collection of objects that we will call the <em>natural numbers</em>. Here, we place ourselves in the context of <a href="https://en.wikipedia.org/wiki/First-order_logic">first-order logic</a>. Logic is not the main topic here, so I will just assume that I have access to some quantifiers, to some predicates, to some variables, and, most importantly, to a relation <span class="math inline">\(=\)</span> which is reflexive, symmetric, transitive, and closed over the natural numbers.</p> <p>The purpose of the axioms is to define a collection of objects that we will call the <em>natural numbers</em>. Here, we place ourselves in the context of <a href="https://en.wikipedia.org/wiki/First-order_logic">first-order logic</a>. Logic is not the main topic here, so I will just assume that I have access to some quantifiers, to some predicates, to some variables, and, most importantly, to a relation <span class="math inline">\(=\)</span> which is reflexive, symmetric, transitive, and closed over the natural numbers.</p>
<p>Without further digressions, let us define two symbols <span class="math inline">\(0\)</span> and <span class="math inline">\(s\)</span> (called <em>successor</em>) such that:</p> <p>Without further digressions, let us define two symbols <span class="math inline">\(0\)</span> and <span class="math inline">\(s\)</span> (called <em>successor</em>) such that:</p>
@ -153,16 +153,22 @@ then <span class="math inline">\(\varphi(n)\)</span> is true for every natural n
<h1 id="going-further">Going further</h1> <h1 id="going-further">Going further</h1>
<p>We have imbued our newly created set of natural numbers with a significant algebraic structure. From there, similar arguments will create more structure, notably by introducing another operation <span class="math inline">\(\times\)</span>, and an order <span class="math inline">\(\leq\)</span>.</p> <p>We have imbued our newly created set of natural numbers with a significant algebraic structure. From there, similar arguments will create more structure, notably by introducing another operation <span class="math inline">\(\times\)</span>, and an order <span class="math inline">\(\leq\)</span>.</p>
<p>It is now a matter of conventional mathematics to construct the integers <span class="math inline">\(\mathbb{Z}\)</span> and the rationals <span class="math inline">\(\mathbb{Q}\)</span> (using equivalence classes), and eventually the real numbers <span class="math inline">\(\mathbb{R}\)</span>.</p> <p>It is now a matter of conventional mathematics to construct the integers <span class="math inline">\(\mathbb{Z}\)</span> and the rationals <span class="math inline">\(\mathbb{Q}\)</span> (using equivalence classes), and eventually the real numbers <span class="math inline">\(\mathbb{R}\)</span>.</p>
<p>It is remarkable how very few (and very simple, as far as you would consider the induction axiom “simple”) axioms are enough to build an entire theory of mathematics. This sort of things makes me agree with Eugene Wigner <a href="#ref-3">(3)</a> when he says that “mathematics is the science of skillful operations with concepts and rules invented just for this purpose”. We drew some arbitrary rules out of thin air, and derived countless properties and theorems from them, basically for our own enjoyment. (As Wigner would say, it is <em>incredible</em> that any of these fanciful inventions coming out of nowhere turned out to be even remotely useful.) Mathematics is done mainly for the mathematician’s own pleasure!</p> <p>It is remarkable how very few (and very simple, as far as you would consider the induction axiom “simple”) axioms are enough to build an entire theory of mathematics. This sort of things makes me agree with Eugene Wigner <span class="citation" data-cites="wignerUnreasonableEffectivenessMathematics1990">(Wigner 1990)</span> when he says that “mathematics is the science of skillful operations with concepts and rules invented just for this purpose”. We drew some arbitrary rules out of thin air, and derived countless properties and theorems from them, basically for our own enjoyment. (As Wigner would say, it is <em>incredible</em> that any of these fanciful inventions coming out of nowhere turned out to be even remotely useful.) Mathematics is done mainly for the mathematician’s own pleasure!</p>
<blockquote> <blockquote>
<p>Mathematics cannot be defined without acknowledging its most obvious feature: namely, that it is interesting — M. Polanyi <a href="#ref-3">(3)</a></p> <p>Mathematics cannot be defined without acknowledging its most obvious feature: namely, that it is interesting — M. Polanyi <span class="citation" data-cites="wignerUnreasonableEffectivenessMathematics1990">(Wigner 1990)</span></p>
</blockquote> </blockquote>
<h1 id="references">References</h1> <h1 id="references" class="unnumbered">References</h1>
<ol> <div id="refs" class="references">
<li><span id="ref-1"></span>Awodey, Steve. Category Theory. 2nd ed. Oxford Logic Guides 52. Oxford ; New York: Oxford University Press, 2010.</li> <div id="ref-awodeyCategoryTheory2010">
<li><span id="ref-2"></span>Gowers, Timothy, June Barrow-Green, and Imre Leader. The Princeton Companion to Mathematics. Princeton University Press, 2010.</li> <p>Awodey, Steve. 2010. <em>Category Theory</em>. 2nd ed. Oxford Logic Guides 52. Oxford ; New York: Oxford University Press.</p>
<li><span id="ref-3"></span>Wigner, Eugene P. ‘The Unreasonable Effectiveness of Mathematics in the Natural Sciences’. In Mathematics and Science, by Ronald E Mickens, 291–306. World Scientific, 1990. <a href="https://doi.org/10.1142/9789814503488_0018" class="uri">https://doi.org/10.1142/9789814503488_0018</a>.</li> </div>
</ol> <div id="ref-gowersPrincetonCompanionMathematics2010">
<p>Gowers, Timothy, June Barrow-Green, and Imre Leader. 2010. <em>The Princeton Companion to Mathematics</em>. Princeton University Press.</p>
</div>
<div id="ref-wignerUnreasonableEffectivenessMathematics1990">
<p>Wigner, Eugene P. 1990. “The Unreasonable Effectiveness of Mathematics in the Natural Sciences.” In <em>Mathematics and Science</em>, by Ronald E Mickens, 291–306. WORLD SCIENTIFIC. <a href="https://doi.org/10.1142/9789814503488_0018" class="uri">https://doi.org/10.1142/9789814503488_0018</a>.</p>
</div>
</div>
</section> </section>
</article> </article>
]]></description> ]]></description>

41
posts/peano.org
View file

@ -5,9 +5,9 @@ date: 2019-03-18
* Introduction * Introduction
I have recently bought the book /Category Theory/ from Steve I have recently bought the book /Category Theory/ from Steve Awodey
Awodey [[ref-1][(1)]] (which is awesome, but probably the topic for another citep:awodeyCategoryTheory2010 is awesome, but probably the topic
post), and a particular passage excited my curiosity: for another post), and a particular passage excited my curiosity:
#+begin_quote #+begin_quote
Let us begin by distinguishing between the following things: Let us begin by distinguishing between the following things:
@ -54,8 +54,9 @@ date: 2019-03-18
this subject. this subject.
In this post, I will try to share my path through Peano's axioms In this post, I will try to share my path through Peano's axioms
[[ref-2][(2)]], because they are very simple, and it is easy to uncover basic citep:gowersPrincetonCompanionMathematics2010, because they are very
algebraic structure from them. simple, and it is easy to uncover basic algebraic structure from
them.
* The Axioms * The Axioms
@ -188,29 +189,21 @@ date: 2019-03-18
It is remarkable how very few (and very simple, as far as you would It is remarkable how very few (and very simple, as far as you would
consider the induction axiom "simple") axioms are enough to build an consider the induction axiom "simple") axioms are enough to build an
entire theory of mathematics. This sort of things makes me agree entire theory of mathematics. This sort of things makes me agree
with Eugene Wigner [[ref-3][(3)]] when he says that "mathematics is the science with Eugene Wigner
of skillful operations with concepts and rules invented just for citep:wignerUnreasonableEffectivenessMathematics1990 when he says
this purpose". We drew some arbitrary rules out of thin air, and that "mathematics is the science of skillful operations with
derived countless properties and theorems from them, basically for concepts and rules invented just for this purpose". We drew some
our own enjoyment. (As Wigner would say, it is /incredible/ that any arbitrary rules out of thin air, and derived countless properties
of these fanciful inventions coming out of nowhere turned out to be and theorems from them, basically for our own enjoyment. (As Wigner
even remotely useful.) Mathematics is done mainly for the would say, it is /incredible/ that any of these fanciful inventions
mathematician's own pleasure! coming out of nowhere turned out to be even remotely useful.)
Mathematics is done mainly for the mathematician's own pleasure!
#+begin_quote #+begin_quote
Mathematics cannot be defined without acknowledging its most obvious Mathematics cannot be defined without acknowledging its most obvious
feature: namely, that it is interesting --- M. Polanyi [[ref-3][(3)]] feature: namely, that it is interesting --- M. Polanyi
citep:wignerUnreasonableEffectivenessMathematics1990
#+end_quote #+end_quote
* References * References
1. <<ref-1>>Awodey, Steve. Category Theory. 2nd ed. Oxford Logic
Guides 52. Oxford ; New York: Oxford University Press, 2010.
2. <<ref-2>>Gowers, Timothy, June Barrow-Green, and Imre Leader. The
Princeton Companion to Mathematics. Princeton University
Press, 2010.
3. <<ref-3>>Wigner, Eugene P. ‘The Unreasonable Effectiveness of
Mathematics in the Natural Sciences’. In Mathematics and Science,
by Ronald E Mickens, 291–306. World
Scientific, 1990. https://doi.org/10.1142/9789814503488_0018.