From 275e10e3dfa4d778c381c3e4c1be4a49f51cc5d8 Mon Sep 17 00:00:00 2001 From: Dimitri Lozeve Date: Wed, 3 Apr 2019 21:38:33 +0200 Subject: [PATCH] Update references in Peano post --- _site/atom.xml | 26 ++++++++++++++++---------- _site/posts/peano.html | 26 ++++++++++++++++---------- _site/rss.xml | 26 ++++++++++++++++---------- posts/peano.org | 41 +++++++++++++++++------------------------ 4 files changed, 65 insertions(+), 54 deletions(-) diff --git a/_site/atom.xml b/_site/atom.xml index 92b5b07..439101f 100644 --- a/_site/atom.xml +++ b/_site/atom.xml @@ -65,7 +65,7 @@

Introduction

-

I have recently bought the book Category Theory from Steve Awodey (1) (which is awesome, but probably the topic for another post), and a particular passage excited my curiosity:

+

I have recently bought the book Category Theory from Steve Awodey (Awodey 2010) is awesome, but probably the topic for another post), and a particular passage excited my curiosity:

Let us begin by distinguishing between the following things: i. categorical foundations for mathematics, ii. mathematical foundations for category theory.

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.”

@@ -73,7 +73,7 @@

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 \(\mathbf{Set}\) of all sets, and even the category \(\mathbf{Cat}\) 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.

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 Peano axioms and the Zermelo-Fraenkel axioms. Also, I should steer clear of the axiom of choice if I could, because one can do strange things with it, and it is equivalent to many different statements. Finally (and this I knew mainly from Logicomix, I must admit), it is impossible for a set of axioms to be both complete and consistent.

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.

-

In this post, I will try to share my path through Peano’s axioms (2), because they are very simple, and it is easy to uncover basic algebraic structure from them.

+

In this post, I will try to share my path through Peano’s axioms (Gowers, Barrow-Green, and Leader 2010), because they are very simple, and it is easy to uncover basic algebraic structure from them.

The Axioms

The purpose of the axioms is to define a collection of objects that we will call the natural numbers. Here, we place ourselves in the context of first-order logic. 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 \(=\) which is reflexive, symmetric, transitive, and closed over the natural numbers.

Without further digressions, let us define two symbols \(0\) and \(s\) (called successor) such that:

@@ -157,16 +157,22 @@ then \(\varphi(n)\) is true for every natural n

Going further

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 \(\times\), and an order \(\leq\).

It is now a matter of conventional mathematics to construct the integers \(\mathbb{Z}\) and the rationals \(\mathbb{Q}\) (using equivalence classes), and eventually the real numbers \(\mathbb{R}\).

-

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 (3) 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 incredible 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!

+

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 (Wigner 1990) 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 incredible 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!

-

Mathematics cannot be defined without acknowledging its most obvious feature: namely, that it is interesting — M. Polanyi (3)

+

Mathematics cannot be defined without acknowledging its most obvious feature: namely, that it is interesting — M. Polanyi (Wigner 1990)

-

References

-
    -
  1. Awodey, Steve. Category Theory. 2nd ed. Oxford Logic Guides 52. Oxford ; New York: Oxford University Press, 2010.
    2. -
  2. Gowers, Timothy, June Barrow-Green, and Imre Leader. The Princeton Companion to Mathematics. Princeton University Press, 2010.
    3. -
  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.
    4. -
+

References

+
+
+

Awodey, Steve. 2010. Category Theory. 2nd ed. Oxford Logic Guides 52. Oxford ; New York: Oxford University Press.

+
+
+

Gowers, Timothy, June Barrow-Green, and Imre Leader. 2010. The Princeton Companion to Mathematics. Princeton University Press.

+
+
+

Wigner, Eugene P. 1990. “The Unreasonable Effectiveness of Mathematics in the Natural Sciences.” In Mathematics and Science, by Ronald E Mickens, 291–306. WORLD SCIENTIFIC. https://doi.org/10.1142/9789814503488_0018.

+
+
]]> diff --git a/_site/posts/peano.html b/_site/posts/peano.html index 76f56f3..8a54a0b 100644 --- a/_site/posts/peano.html +++ b/_site/posts/peano.html @@ -31,7 +31,7 @@

Introduction

-

I have recently bought the book Category Theory from Steve Awodey (1) (which is awesome, but probably the topic for another post), and a particular passage excited my curiosity:

+

I have recently bought the book Category Theory from Steve Awodey (Awodey 2010) is awesome, but probably the topic for another post), and a particular passage excited my curiosity:

Let us begin by distinguishing between the following things: i. categorical foundations for mathematics, ii. mathematical foundations for category theory.

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.”

@@ -39,7 +39,7 @@

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 \(\mathbf{Set}\) of all sets, and even the category \(\mathbf{Cat}\) 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.

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 Peano axioms and the Zermelo-Fraenkel axioms. Also, I should steer clear of the axiom of choice if I could, because one can do strange things with it, and it is equivalent to many different statements. Finally (and this I knew mainly from Logicomix, I must admit), it is impossible for a set of axioms to be both complete and consistent.

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.

-

In this post, I will try to share my path through Peano’s axioms (2), because they are very simple, and it is easy to uncover basic algebraic structure from them.

+

In this post, I will try to share my path through Peano’s axioms (Gowers, Barrow-Green, and Leader 2010), because they are very simple, and it is easy to uncover basic algebraic structure from them.

The Axioms

The purpose of the axioms is to define a collection of objects that we will call the natural numbers. Here, we place ourselves in the context of first-order logic. 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 \(=\) which is reflexive, symmetric, transitive, and closed over the natural numbers.

Without further digressions, let us define two symbols \(0\) and \(s\) (called successor) such that:

@@ -123,16 +123,22 @@ then \(\varphi(n)\) is true for every natural n

Going further

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 \(\times\), and an order \(\leq\).

It is now a matter of conventional mathematics to construct the integers \(\mathbb{Z}\) and the rationals \(\mathbb{Q}\) (using equivalence classes), and eventually the real numbers \(\mathbb{R}\).

-

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 (3) 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 incredible 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!

+

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 (Wigner 1990) 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 incredible 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!

-

Mathematics cannot be defined without acknowledging its most obvious feature: namely, that it is interesting — M. Polanyi (3)

+

Mathematics cannot be defined without acknowledging its most obvious feature: namely, that it is interesting — M. Polanyi (Wigner 1990)

-

References

-
    -
  1. Awodey, Steve. Category Theory. 2nd ed. Oxford Logic Guides 52. Oxford ; New York: Oxford University Press, 2010.
    2. -
  2. Gowers, Timothy, June Barrow-Green, and Imre Leader. The Princeton Companion to Mathematics. Princeton University Press, 2010.
    3. -
  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.
    4. -
+

References

+
+
+

Awodey, Steve. 2010. Category Theory. 2nd ed. Oxford Logic Guides 52. Oxford ; New York: Oxford University Press.

+
+
+

Gowers, Timothy, June Barrow-Green, and Imre Leader. 2010. The Princeton Companion to Mathematics. Princeton University Press.

+
+
+

Wigner, Eugene P. 1990. “The Unreasonable Effectiveness of Mathematics in the Natural Sciences.” In Mathematics and Science, by Ronald E Mickens, 291–306. WORLD SCIENTIFIC. https://doi.org/10.1142/9789814503488_0018.

+
+
diff --git a/_site/rss.xml b/_site/rss.xml index 5a13610..0b5d858 100644 --- a/_site/rss.xml +++ b/_site/rss.xml @@ -61,7 +61,7 @@

Introduction

-

I have recently bought the book Category Theory from Steve Awodey (1) (which is awesome, but probably the topic for another post), and a particular passage excited my curiosity:

+

I have recently bought the book Category Theory from Steve Awodey (Awodey 2010) is awesome, but probably the topic for another post), and a particular passage excited my curiosity:

Let us begin by distinguishing between the following things: i. categorical foundations for mathematics, ii. mathematical foundations for category theory.

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.”

@@ -69,7 +69,7 @@

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 \(\mathbf{Set}\) of all sets, and even the category \(\mathbf{Cat}\) 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.

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 Peano axioms and the Zermelo-Fraenkel axioms. Also, I should steer clear of the axiom of choice if I could, because one can do strange things with it, and it is equivalent to many different statements. Finally (and this I knew mainly from Logicomix, I must admit), it is impossible for a set of axioms to be both complete and consistent.

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.

-

In this post, I will try to share my path through Peano’s axioms (2), because they are very simple, and it is easy to uncover basic algebraic structure from them.

+

In this post, I will try to share my path through Peano’s axioms (Gowers, Barrow-Green, and Leader 2010), because they are very simple, and it is easy to uncover basic algebraic structure from them.

The Axioms

The purpose of the axioms is to define a collection of objects that we will call the natural numbers. Here, we place ourselves in the context of first-order logic. 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 \(=\) which is reflexive, symmetric, transitive, and closed over the natural numbers.

Without further digressions, let us define two symbols \(0\) and \(s\) (called successor) such that:

@@ -153,16 +153,22 @@ then \(\varphi(n)\) is true for every natural n

Going further

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 \(\times\), and an order \(\leq\).

It is now a matter of conventional mathematics to construct the integers \(\mathbb{Z}\) and the rationals \(\mathbb{Q}\) (using equivalence classes), and eventually the real numbers \(\mathbb{R}\).

-

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 (3) 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 incredible 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!

+

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 (Wigner 1990) 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 incredible 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!

-

Mathematics cannot be defined without acknowledging its most obvious feature: namely, that it is interesting — M. Polanyi (3)

+

Mathematics cannot be defined without acknowledging its most obvious feature: namely, that it is interesting — M. Polanyi (Wigner 1990)

-

References

-
    -
  1. Awodey, Steve. Category Theory. 2nd ed. Oxford Logic Guides 52. Oxford ; New York: Oxford University Press, 2010.
    2. -
  2. Gowers, Timothy, June Barrow-Green, and Imre Leader. The Princeton Companion to Mathematics. Princeton University Press, 2010.
    3. -
  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.
    4. -
+

References

+
+
+

Awodey, Steve. 2010. Category Theory. 2nd ed. Oxford Logic Guides 52. Oxford ; New York: Oxford University Press.

+
+
+

Gowers, Timothy, June Barrow-Green, and Imre Leader. 2010. The Princeton Companion to Mathematics. Princeton University Press.

+
+
+

Wigner, Eugene P. 1990. “The Unreasonable Effectiveness of Mathematics in the Natural Sciences.” In Mathematics and Science, by Ronald E Mickens, 291–306. WORLD SCIENTIFIC. https://doi.org/10.1142/9789814503488_0018.

+
+
]]> diff --git a/posts/peano.org b/posts/peano.org index 766e609..d9e689d 100644 --- a/posts/peano.org +++ b/posts/peano.org @@ -5,9 +5,9 @@ date: 2019-03-18 * Introduction - I have recently bought the book /Category Theory/ from Steve - Awodey [[ref-1][(1)]] (which is awesome, but probably the topic for another - post), and a particular passage excited my curiosity: + I have recently bought the book /Category Theory/ from Steve Awodey + citep:awodeyCategoryTheory2010 is awesome, but probably the topic + for another post), and a particular passage excited my curiosity: #+begin_quote Let us begin by distinguishing between the following things: @@ -54,8 +54,9 @@ date: 2019-03-18 this subject. 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 - algebraic structure from them. + citep:gowersPrincetonCompanionMathematics2010, because they are very + simple, and it is easy to uncover basic algebraic structure from + them. * The Axioms @@ -188,29 +189,21 @@ date: 2019-03-18 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 [[ref-3][(3)]] 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 /incredible/ 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! + with Eugene Wigner + citep:wignerUnreasonableEffectivenessMathematics1990 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 /incredible/ 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! #+begin_quote 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 * References -1. <>Awodey, Steve. Category Theory. 2nd ed. Oxford Logic - Guides 52. Oxford ; New York: Oxford University Press, 2010. -2. <>Gowers, Timothy, June Barrow-Green, and Imre Leader. The - Princeton Companion to Mathematics. Princeton University - Press, 2010. -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. -