Mathematical Philosophy (10 Mar 2003)

http://suo.ieee.org/email/msg09053.html

SUO: Re: Languages and Efficiency

To: "John F. Sowa" <sowa@bestweb.net> 
Subject: SUO: Re: Languages and Efficiency 
From: Jon Awbrey <jawbrey@oakland.edu> 
Date: Mon, 10 Mar 2003 14:14:07 -0500 
CC: standard-upper-ontology@ieee.org 
References: <3E6CA6C3.5090802@bestweb.net> 
Reply-To: Jon Awbrey <jawbrey@oakland.edu> 
Sender: owner-standard-upper-ontology@majordomo.ieee.org

John F. Sowa wrote:
> 
> For over 20 years, there has been a vocal community
> that has been popularizing the idea that knowledge
> representation languages should be restricted in
> expressive power to limited subsets of logic.

John,

As it happens, I have been concerned of late to trace a not unrelated
brace of misconceptions back to their source -- I give you one guess:

| Mathematics and logic, historically speaking, have been entirely
| distinct studies.  Mathematics has been connected with science,
| logic with Greek.  But both have developed in modern times:
| logic has become more mathematical and mathematics has
| become more logical.  The consequence is that it has
| now become wholly impossible to draw a line between
| the two;  in fact, the two are one.  They differ as
| boy and man:  logic is the youth of mathematics and
| mathematics is the manhood of logic.  This view is
| resented by logicians who, having spent their time
| in the study of classical texts, are incapable of
| following a piece of symbolic reasoning, and by
| mathematicians who have learnt a technique
| without troubling to inquire into its
| meaning or justification.  Both types
| are now fortunately growing rarer.
| So much of modern mathematical work
| is obviously on the border-line of
| logic, so much of modern logic is
| symbolic and formal, that the very
| close relationship of logic and
| mathematics has become obvious
| to every instructed student.
| The proof of their identity is,
| of course, a matter of detail:
| starting with premisses which
| would be universally admitted
| to belong to logic, and arriving
| by deduction at results which as
| obviously belong to mathematics,
| we find that there is no point
| at which a sharp line can be drawn,
| with logic to the left and mathematics
| to the right.  If there are still those
| who do not admit the identity of logic and
| mathematics, we may challenge them to indicate
| at what point, in the successive definitions and
| deductions of 'Principia Mathematica', they consider
| that logic ends and mathematics begins.  It will then
| be obvious that any answer must be quite arbitrary.
|
| Russell, IMP, pages 194-195.
|
| Bertrand Russell, 'Introduction to Mathematical Philosophy',
| Routledge, London, UK, 2000.  Originally published in 1919.

The fact is, and the fact has been for a long time now, that all of our
working models of reality take on a substantial mathematical form, but
the sadder fact is that the mainstream logicians of the 1900's never
got as far as the mainstream mathematics of the 1800's, and so they
had a tendency to say looney things like Russell said just above.
In spite of the self-vitiating circularity of his reasoning,
and in spite of the fact that Goedel took up his challenge
and demolished his position, we are still suffering from
the hangers-on-&-over of this "terrible identity" that:
Mathematics = Logic = What Russell Knew About Logic.

Jon Awbrey

Philosophy of Mathematics (18 May 2006)

Philosophy of mathematics is a branch of philosophy that addresses questions such as (1) What is the nature of mathematics, including mathematical inquiry, knowledge, and truth, and (2) How do mathematical models serve in describing empirical phenomena?

Relation to mathematics proper

Linguists, logicians, psychologists, and philosophers have put forward explanations of where mathematics comes from, what it's about, and how it ought to be done. That being settled, the question arises as to where the philosophy of mathematics comes from.

From the writings that have come down to us, it appears that the philosophy of mathematics and the practice of mathematics went hand in hand for most of human history, with the same people engaged in both aspects of a single activity, the natural emphasis being on the practical side, but by its very nature demanding considerable examination of alternative ways that it might be done better. What little external critique there was appears to have come mainly from the great comic writers like Aristophanes. That makes for a likely story, as seen from a distance, but it is more likely an artifact of the sample of works that are extant.

The record grows clearer and more detailed with the advents of the Renaissance and the Enlightenment that the ways of mathematics and science in general were beginning to attract the attention of artisans and astute thinkers from areas beyond the practice of mathematics proper.

It is necessary at the outset to distinguish the philosophy of mathematics from the philosophy of mathematicians. Philosophy of mathematics has its source in any moment that a person reflects on mathematical practice, whether it is another person's practice or that person's own. Having made that distinction between the more generic reflection on mathematics and its more specialized reflexive application, it is possible to see yet another distinction, analogous to what medieval logicians call logica docens, logic as taught, and logica utens, logic as used. C.S. Peirce, as a logician, mathematician, and philosopher who found it useful to study the past on the way to creating the future of mathematics, is a useful source here:

But mathematics performs its reasoning by a logica utens which it develops for itself, and has no need of any appeal to a logica docens; for no disputes about reasoning arise in mathematics which need to be submitted to the principles of the philosophy of thought for decision. (C.S. Peirce, CP 1.417).

Peirce is here signing on to a declaration of independence for mathematics that he knows is nothing new, that many others have signed on before, but it remains a steadfast position that he affirms on many occasions and in many different ways all throughout his work on the foundations of mathematics.

Relation to philosophy proper

The terms philosophy of mathematics and mathematical philosophy are not synonyms. The latter is more often used to mean at least three distinct things. One sense refers to a project of formalizing a philosophical subject matter, say, aesthetics, ethics, metaphysics, or theology, in a purportedly more exact and rigorous form, as for example the labors of Scholastic theologicians, or the systematic aims of Leibniz and Spinoza. Another sense refers to the working philosophy of an individual practitioner or a like-minded community of practicing mathematicians. Apart from that, some philosophers understand the term mathematical philosophy as an allusion to the approach taken by Bertrand Russell in his book Introduction to Mathematical Philosophy.

Some philosophers of mathematics view their task as giving an account of mathematics and mathematical practice as it stands, as interpretation rather than criticism. Criticisms can, however, have important ramifications for mathematical practice, so the philosophy of mathematics can be of direct interest to working mathematicians, particularly in new fields where the process of peer review of mathematical proofs is not firmly established, raising the probability of an undetected error. Such errors can thus only be reduced by knowing where they are likely to arise. This is a prime concern of the philosophy of mathematics.

Philosophy of Mathematics (20 May 2006)

Philosophy of mathematics is a branch of philosophy that addresses questions such as (1) What is the nature of mathematics, including mathematical inquiry, knowledge, and truth, (2) Why do mathematical models work so well in describing empirical phenomena?, (3) What are the limitations of mathemetics in describing said empirical phenomena (i.e. do they work well?), and (3) What is the sense of aesthetics inherent in mathematical practice?

Relation to philosophy proper

Philosophy of mathematics today is a standard topic in university curricula, pursued by professional philosophers who specialize in that area. But that is a relatively recent development, and even today philosophy of mathematics is not just an entry in a college course catalogue — it is an intellectual activity and a cultural resource to which a diversity of thinkers down through the ages have contributed their ideas on the conduct of mathematical inquiry and the nature of mathematical knowledge. On the current scene, many philosophers of mathematics aim to give accounts of this form of inquiry and its products as they stand, while others emphasize a role for themselves that goes beyond simple interpretation to critical analysis. Critiques of mathematical reasoning that stem from a lack of familiarity with actual practice, however, tend to be ignored by mathematicians, and thus have little impact on mathematical progress.

The terms philosophy of mathematics and mathematical philosophy are synonyms to some but signify different intentions to others. The important thing is not the labels themselves but the two different approaches to philosophical reflection on mathematical practice that are marked by some while remaining a matter of indifference to others. The difference has to do with two distinct ways of viewing the relation between mathematical practice and philosophical reflection. The critical question is whether philosophy grows out of reflection on practice, or not.

Relation to mathematics proper

The question arises as to where the philosophy of mathematics comes from. What are its incitements, its occasions, its motive springs and catches? The question applies at many different horizons and scopes of interest. It can be asked as a question of historical genesis within the global context of a particular culture, it can be asked as a question of intellectual development within more local and transient communities of inquiry, or it can be asked with respect to the individual lives of inquiry that have fed the stream of thought that we now call the philosophy of mathematics.

At the cultural level, the record suggests that the philosophy of mathematics and the practice of mathematics went hand in hand for most of human history, with the same people engaged in both aspects of an integral activity. Though there was a natural emphasis at first on the practical side of mathematical work, its very nature demands a considerable examination of alternative ways that it might be done better. At this stage of the game, the discipline of mathematics would have involved mostly internal forms of criticism. What little external critique there was appears to have come mainly from the great comic writers like Aristophanes.

With the Renaissance and the Enlightenment, mathematics, and science in general, began to attract the attention of thinkers from areas beyond the practice of mathematics proper.

At the individual level, it is necessary to distinguish the philosophy of mathematics from the philosophy of mathematicians. Philosophy of mathematics has its source in any moment that a person reflects on mathematical practice, whether it is another person's practice or one's own. Having made that distinction, it is possible to see a further one, analogous to what logicians call logica docens, logic as taught, and logica utens, logic as used.

Philosophy of Mathematics (21 May 2006)

Philosophy of mathematics is a form of philosophical inquiry that examines the record of mathematical inquiry and and poses questions regarding its aims, its conduct, and its results. Although the questions are diverse and never-ending, a number of recurrent themes can be recognized:

What are the sources of mathematical subject matter?
What does it mean to refer to a mathematical object?
What is the character of a mathematical proposition?
What kinds of inquiry play a role in mathematics?
What is the objective of mathematical inquiry?
What gives mathematics its grip on experience?

A quick run through this slate of questions, touching on a sample of the answers that have been given so far, makes for a ready introduction to the philosophy of mathematics.

Overview

Philosophy of mathematics today is a standard topic in university curricula, pursued by professional philosophers who specialize in that area. Many thinkers down through the ages have contributed their ideas concerning the conduct of mathematical inquiry and the nature of mathematical objects and knowledge. Today, many philosophers of mathematics aim to give accounts of this form of inquiry and its products as they stand, while others emphasize a role for themselves that goes beyond simple interpretation to critical analysis.

The terms philosophy of mathematics and mathematical philosophy are equivalent to some but the latter term conveys a wider range of meaning to others. A discusssion of mathematical philosophy among working mathematicians refers to the guiding principles of the work, both disciplinary norms and personal variations. Another use of the latter term refers to a project of formalizing a philosophical subject matter, say, aesthetics, ethics, logic, metaphysics, or theology, in a putatively more exact and rigorous form. One thinks here of the dedicated labors of Scholastic theologians or the systematic programmes of Leibniz and Spinoza. Apart from that, a few philosophers understand the term mathematical philosophy as alluding to the approach taken by Bertrand Russell in his Introduction to Mathematical Philosophy (1919).

Schools of thought

As the 20th century progressed, however, philosophical opinions diverged as to just how well-founded were the questions about foundations that were raised at its opening (Putnam 1967).

Philosophy of mathematics (22 May 2006 a)

Philosophy of mathematics is the branch of philosophy that considers questions regarding the nature and practice of mathematics. Issues addressed include the nature of the reality of mathematical objects, why mathematical models seem to work so well in describing empirical phenomena and what the limitations of those models are, issues of proof, aesthetics, and mathematical beauty, the relation of formal logic to mathematics, and other questions.

Context

Many thinkers down through the ages have contributed their ideas concerning the conduct of mathematical inquiry and the nature of mathematical objects and knowledge. Today, many philosophers of mathematics aim to give accounts of this form of inquiry and its products as they stand, while others emphasize a role for themselves that goes beyond simple interpretation to critical analysis.

The terms philosophy of mathematics and mathematical philosophy are often taken to be synonomous, but others distinguish between them. The latter may be used to mean at least three distinct things. One sense refers to a project of formalizing a philosophical subject matter, say, aesthetics, ethics, metaphysics, or theology, in a purportedly more exact and rigorous form, as for example the labors of Scholastic theologians, or the systematic aims of Leibniz and Spinoza. Another sense refers to the working philosophy of an individual practitioner or a like-minded community of practicing mathematicians. Additionally, some understand the term mathematical philosophy to be an allusion to the approach taken by Bertrand Russell in his book Introduction to Mathematical Philosophy.

Schools of thought

The philosophy of mathematics has seen several different schools, distinguished by their pictures of mathematical metaphysics and epistemology. Important questions these schools have considered include, "What are mathematical objects?" and "How and why does mathematics work?". Other important questions include "How do we know mathematical truths?" and "What do statements about mathematics mean?" as well as questions concerning logic and the foundations of mathematics, which is currently the topic of greatest interest to philosophers of mathematics.

Three schools, intuitionism, logicism, and formalism, emerged around the start of the 20th century in response to the increasingly widespread realisation that mathematics (as it stood), and analysis in particular, did not live up to the standards of certainty and rigour with which it had traditionally been credited. Each school addresses the issues that came to the fore at that time, either attempting to resolve them or claiming that mathematics is not entitled to its status as our most trusted knowledge.

Core concepts such as axioms, order, and set were given renewed emphasis. Finally, a subfield of mathematics and/or philosophy that might be called metamathematics emerged due to the newly recognized need for a mathematical rigorization of the very methods and assumptions of mathematical reasoning.

Surprising and counterintuitive developments in formal logic and set theory early in the 20th century led to new questions concerning what was traditionally called the foundations of mathematics. As the century unfolded, the initial focus of concern expanded into an open exploration of the axiomatic basis of set theory, which has been taken for granted by many as the universal medium for mathematical inquiry. In mathematics as in physics, however, the age of all-pervading media was already waning. At the midpoint of the century, mathematical category theory arose as a new contender for the natural language of mathematical thinking (Mac Lane 1998). The specialty known as metamathematics emerged to address the task of more rigorously formalizing the very methods of mathematical reasoning. As the 20th century progressed, however, philosophical opinions diverged as to just how well-founded were the questions about foundations that were raised at its opening. Hilary Putnam (1967) summed up one common view of the situation in the last third of the century by saying:

When philosophy discovers something wrong with science, sometimes science has to be changed — Russell's paradox comes to mind, as does Berkeley's attack on the actual infinitesimal — but more often it is philosophy that has to be changed. I do not think that the difficulties that philosophy finds with classical mathematics today are genuine difficulties; and I think that the philosophical interpretations of mathematics that we are being offered on every hand are wrong, and that 'philosophical interpretation' is just what mathematics doesn't need. (Putnam 1967/1996, 169–170).

The schools are addressed separately here and their assumptions explained:

Philosophy of mathematics (22 May 2006 b)

Philosophy of mathematics is the branch of philosophy that considers questions regarding the nature and practice of mathematics. Issues addressed include the nature of the reality of mathematical objects, why mathematical models seem to work so well in describing empirical phenomena and what the limitations of those models are, issues of proof, aesthetics, and mathematical beauty, the relation of formal logic to mathematics, and other questions.

Context

Many thinkers down through the ages have contributed their ideas concerning the conduct of mathematical inquiry and the nature of mathematical objects and knowledge. Today, many philosophers of mathematics aim to give accounts of this form of inquiry and its products as they stand, while others emphasize a role for themselves that goes beyond simple interpretation to critical analysis.

The terms philosophy of mathematics and mathematical philosophy are often taken to be synonomous, but others distinguish between them. The latter may be used to mean at least three distinct things. One sense refers to a project of formalizing a philosophical subject matter, say, aesthetics, ethics, metaphysics, or theology, in a purportedly more exact and rigorous form, as for example the labors of Scholastic theologians, or the systematic aims of Leibniz and Spinoza. Another sense refers to the working philosophy of an individual practitioner or a like-minded community of practicing mathematicians. Additionally, some understand the term mathematical philosophy to be an allusion to the approach taken by Bertrand Russell in his book Introduction to Mathematical Philosophy.

Schools of thought

The philosophy of mathematics has seen several different schools, distinguished by their pictures of mathematical metaphysics and epistemology. Important questions these schools have considered include, "What are mathematical objects?" and "How and why does mathematics work?". Other important questions include "How do we know mathematical truths?" and "What do statements about mathematics mean?" as well as questions concerning logic and the foundations of mathematics, which is currently the topic of greatest interest to philosophers of mathematics.

Three schools, intuitionism, logicism, and formalism, emerged around the start of the 20th century in response to the increasingly widespread realisation that mathematics (as it stood), and analysis in particular, did not live up to the standards of certainty and rigour with which it had traditionally been credited. Each school addresses the issues that came to the fore at that time, either attempting to resolve them or claiming that mathematics is not entitled to its status as our most trusted knowledge.

Core concepts such as axioms, order, and set were given renewed emphasis. Finally, a subfield of mathematics and/or philosophy that might be called metamathematics emerged due to the newly recognized need for a mathematical rigorization of the very methods and assumptions of mathematical reasoning.

Surprising and counterintuitive developments in formal logic and set theory early in the 20th century led to new questions concerning what was traditionally called the foundations of mathematics. As the century unfolded, the initial focus of concern expanded into an open exploration of the axiomatic basis of set theory, which has been taken for granted by many as the universal medium for mathematical inquiry. In mathematics as in physics, however, the age of all-pervading media was already waning. At the midpoint of the century, mathematical category theory arose as a new contender for the natural language of mathematical thinking (Mac Lane 1998). The specialty known as metamathematics emerged to address the task of more rigorously formalizing the very methods of mathematical reasoning. As the 20th century progressed, however, philosophical opinions diverged as to just how well-founded were the questions about foundations that were raised at its opening. Hilary Putnam (1967) summed up one common view of the situation in the last third of the century by saying:

When philosophy discovers something wrong with science, sometimes science has to be changed — Russell's paradox comes to mind, as does Berkeley's attack on the actual infinitesimal — but more often it is philosophy that has to be changed. I do not think that the difficulties that philosophy finds with classical mathematics today are genuine difficulties; and I think that the philosophical interpretations of mathematics that we are being offered on every hand are wrong, and that 'philosophical interpretation' is just what mathematics doesn't need. (Putnam 1967/1996, 169–170).

The schools are addressed separately here and their assumptions explained:

Philosophy of mathematics (23 May 2006)

Philosophy of mathematics is the branch of philosophy that considers questions regarding the nature and practice of mathematics. Issues addressed include the nature of the reality of mathematical objects, why mathematical models seem to work so well in describing empirical phenomena and what the limitations of those models are, issues of proof, aesthetics, and mathematical beauty, the relation of formal logic to mathematics, and other questions.

Context

Many thinkers down through the ages have contributed their ideas concerning the conduct of mathematical inquiry and the nature of mathematical objects and knowledge. Today, many philosophers of mathematics aim to give accounts of this form of inquiry and its products as they stand, while others emphasize a role for themselves that goes beyond simple interpretation to critical analysis.

The terms philosophy of mathematics and mathematical philosophy are often taken to be synonymous, but others distinguish between them. The latter may be used to mean at least three distinct things. One sense refers to a project of formalizing a philosophical subject matter, say, aesthetics, ethics, logic, metaphysics, or theology, in a purportedly more exact and rigorous form, as for example the labors of Scholastic theologians, or the systematic aims of Leibniz and Spinoza. Another sense refers to the working philosophy of an individual practitioner or a like-minded community of practicing mathematicians. Additionally, some understand the term mathematical philosophy to be an allusion to the approach taken by Bertrand Russell in his book Introduction to Mathematical Philosophy.

Schools of thought

The philosophy of mathematics has seen several different schools, distinguished by their pictures of mathematical metaphysics and epistemology. Important questions these schools have considered include, "What are mathematical objects?" and "How and why does mathematics work?". Other important questions include "How do we know mathematical truths?" and "What do statements about mathematics mean?" as well as questions concerning logic and the foundations of mathematics, which is currently the topic of greatest interest to philosophers of mathematics.

Three schools, intuitionism, logicism, and formalism, emerged around the start of the 20th century in response to the increasingly widespread realisation that mathematics (as it stood), and analysis in particular, did not live up to the standards of certainty and rigor with which it had traditionally been credited. Each school addresses the issues that came to the fore at that time, either attempting to resolve them or claiming that mathematics is not entitled to its status as our most trusted knowledge.

Surprising and counterintuitive developments in formal logic and set theory early in the 20th century led to new questions concerning what was traditionally called the foundations of mathematics. The anomalies of the intuition that the turn of the century turned up, and with regard to such unusual suspects as the formerly "simple" idea of a set, are well-represented in the following remark of Julius König.

That the word "set" is being used indiscriminately for completely different notions and that this is the source of the apparent paradoxes of this young branch of science, that, moreover, set theory itself can no more dispense with axiomatic assumptions than can any other exact science and that these assumptions, just as in other disciplines, are subject to a certain arbitrariness, even if they lie much deeper here — I do not want to represent any of this as something new. (König 1905, 145).

As the century unfolded, the initial focus of concern expanded to an open exploration of the fundamental axioms of mathematics, the axiomatic approach having been taken for granted for millenia as the natural basis for mathematics. In mathematics as in physics, new and unexpected ideas had arisen and significant changes were coming. Gradually, an interdisciplinary field between mathematics and philosophy that came to be called metamathematics emerged to address the task of more rigorously formalizing the very methods of mathematical reasoning. Core concepts such as axiom, order, and set were given renewed emphasis.

At the midpoint of the century, mathematical category theory arose as a new contender for the natural language of mathematical thinking (Mac Lane 1998). As the 20th century progressed, however, philosophical opinions diverged as to just how well-founded were the questions about foundations that were raised at its opening. Hilary Putnam (1967) summed up one common view of the situation in the last third of the century by saying:

When philosophy discovers something wrong with science, sometimes science has to be changed — Russell's paradox comes to mind, as does Berkeley's attack on the actual infinitesimal — but more often it is philosophy that has to be changed. I do not think that the difficulties that philosophy finds with classical mathematics today are genuine difficulties; and I think that the philosophical interpretations of mathematics that we are being offered on every hand are wrong, and that 'philosophical interpretation' is just what mathematics doesn't need. (Putnam 1967/1996, 169–170).

Philosophy of mathematics today proceeds along several different lines of inquiry, by philosophers of mathematics, logicians, and mathematicians, and there are many schools of thought on the subject. The schools are addressed separately below, and their assumptions explained:

Philosophy of mathematics (24 May 2006 a)

Philosophy of mathematics is the branch of philosophy that considers questions regarding the nature and practice of mathematics. Issues addressed include the nature of the reality of mathematical objects, why mathematical models seem to work so well in describing empirical phenomena and what the limitations of those models are, issues of proof, aesthetics, and mathematical beauty, the relation of formal logic to mathematics, and other questions.

Context

Many thinkers down through the ages have contributed their ideas concerning the conduct of mathematical inquiry and the nature of mathematical objects and knowledge. Today, many philosophers of mathematics aim to give accounts of this form of inquiry and its products as they stand, while others emphasize a role for themselves that goes beyond simple interpretation to critical analysis.

The terms philosophy of mathematics and mathematical philosophy are often taken to be synonymous, but others distinguish between them. The latter may be used to mean at least three distinct things. One sense refers to a project of formalizing a philosophical subject matter, say, aesthetics, ethics, logic, metaphysics, or theology, in a purportedly more exact and rigorous form, as for example the labors of Scholastic theologians, or the systematic aims of Leibniz and Spinoza. Another sense refers to the working philosophy of an individual practitioner or a like-minded community of practicing mathematicians. Additionally, some understand the term mathematical philosophy to be an allusion to the approach taken by Bertrand Russell in his book Introduction to Mathematical Philosophy.

Historical overview

Philosophy of mathematics in the 20th century

Philosophical reflection on mathematics naturally brings to light a very broad slate of questions. One of the most salient concerns the relation between logic and mathematics at their respective foundations, historically among the topics of greatest interest to philosophers of mathematics. More concrete questions that typically arise are "How and why does mathematics work?" and "What does it mean to speak of a mathematical object?" Other important questions include "What is the character of a mathematical proposition?", "How do we come to know a mathematical truth?", and "What do we know when we know it?"

At the start of the 20th century, philosophers of mathematics were already beginning to divide into various schools of thought about these and many other questions, broadly distinguished by their pictures of mathematical epistemology and ontology. Three schools, formalism, intuitionism, and logicism, emerged at this time, partly in response to the increasingly widespread worry that mathematics as it stood, and analysis in particular, did not live up to the standards of certainty and rigor that had been taken for granted. Each school addressed the issues that came to the fore at that time, either attempting to resolve them or claiming that mathematics is not entitled to its status as our most trusted knowledge.

Surprising and counterintuitive developments in formal logic and set theory early in the 20th century led to new questions concerning what was traditionally called the foundations of mathematics. Conceptual anomalies began to appear in what had been considered the fairly secure territories of simple intuitions, for instance, those about classes, collections, or sets. The tenor of the times is well conveyed by the following remark of Julius König.

That the word "set" is being used indiscriminately for completely different notions and that this is the source of the apparent paradoxes of this young branch of science, that, moreover, set theory itself can no more dispense with axiomatic assumptions than can any other exact science and that these assumptions, just as in other disciplines, are subject to a certain arbitrariness, even if they lie much deeper here — I do not want to represent any of this as something new. (König 1905, 145).

As the century unfolded, the initial focus of concern expanded to an open exploration of the fundamental axioms of mathematics, the axiomatic approach having been taken for granted for millenia as the natural basis for mathematics. In mathematics as in physics, new and unexpected ideas had arisen and significant changes were coming. Gradually, an interdisciplinary field between mathematics and philosophy that came to be called metamathematics emerged to address the task of more rigorously formalizing the very methods of mathematical reasoning. Core concepts such as axiom, order, and set were given renewed emphasis.

At the midpoint of the century, mathematical category theory arose as a new contender for the natural language of mathematical thinking (Mac Lane 1998). As the 20th century entered retirement, philosophical opinions diverged once again, with a sugnificant contingent beginning to reconsider just how well founded were the questions about foundations that were raised at its opening. Hilary Putnam (1967) summed up one common view of the situation in the last third of the century by saying:

When philosophy discovers something wrong with science, sometimes science has to be changed — Russell's paradox comes to mind, as does Berkeley's attack on the actual infinitesimal — but more often it is philosophy that has to be changed. I do not think that the difficulties that philosophy finds with classical mathematics today are genuine difficulties; and I think that the philosophical interpretations of mathematics that we are being offered on every hand are wrong, and that 'philosophical interpretation' is just what mathematics doesn't need. (Putnam 1967/1996, 169–170).

Philosophy of mathematics (24 May 2006 b)

Philosophy of mathematics is a philosophical inquiry into the aims, the context, the practice, and the results of mathematical inquiry. Among the focal questions that are encountered in the history and literature of the subject there are found the following recurrent themes:

What are the sources of mathematical subject matter?
What does it mean to refer to a mathematical object?
What is the character of a mathematical proposition?
What kinds of inquiry play a role in mathematics?
What are the objectives of mathematical inquiry?
What gives mathematics its hold on experience?
What links mathematical beauty to mathematical truth?

A quick run through this slate of questions, touching on a variety of the answers that have been given up to the present time, makes for a ready introduction to the philosophy of mathematics.

Philosophy of mathematics (25 May 2006 a)

Philosophy of mathematics is the branch of philosophy that studies the philosophical assumptions, foundations, and implications of mathematics. Recurrent themes include:

What topics are in the scope of mathematics?
What is the nature of mathematical objects?
What is a proper mathematical proof?
What are the fundamental assumptions or axioms used in mathematics?
What is the relationship between mathematics and logic?
What is the reason that mathematics is useful in the sciences?
What is mathematical beauty and truth?

This article focuses on the Western philosophy of mathematics.

Context

Many thinkers down through the ages have contributed their ideas concerning the conduct of mathematical inquiry and the nature of mathematical objects and knowledge. Today, many philosophers of mathematics aim to give accounts of this form of inquiry and its products as they stand, while others emphasize a role for themselves that goes beyond simple interpretation to critical analysis.

The terms philosophy of mathematics and mathematical philosophy are often taken to be synonymous, but others distinguish between them. The latter may be used to mean at least three distinct things. One sense refers to a project of formalizing a philosophical subject matter, say, aesthetics, ethics, logic, metaphysics, or theology, in a purportedly more exact and rigorous form, as for example the labors of Scholastic theologians, or the systematic aims of Leibniz and Spinoza. Another sense refers to the working philosophy of an individual practitioner or a like-minded community of practicing mathematicians. Additionally, some understand the term mathematical philosophy to be an allusion to the approach taken by Bertrand Russell in his book Introduction to Mathematical Philosophy.

Historical overview

Philosophy of mathematics in the 20th century

Philosophical reflection on mathematics naturally brings to light a very broad array of questions. One of the most salient issues, historically a topic of great interest to the philosophers of mathematics, concerns the relation between logic and mathematics at their joint foundations. More concrete questions that typically arise are "How and why does mathematics work?" and "What does it mean to speak of a mathematical object?" Other important questions include "What is the character of a mathematical proposition?", "How do we come to know a mathematical truth?", and "What do we know when we know it?"

At the start of the 20th century, philosophers of mathematics were already beginning to divide into various schools of thought about these and many other questions, broadly distinguished by their pictures of mathematical epistemology and ontology. Three schools, formalism, intuitionism, and logicism, emerged at this time, partly in response to the increasingly widespread worry that mathematics as it stood, and analysis in particular, did not live up to the standards of certainty and rigor that had been taken for granted. Each school addressed the issues that came to the fore at that time, either attempting to resolve them or claiming that mathematics is not entitled to its status as our most trusted knowledge.

Surprising and counterintuitive developments in formal logic and set theory early in the 20th century led to new questions concerning what was traditionally called the foundations of mathematics. As the century unfolded, the initial focus of concern expanded to an open exploration of the fundamental axioms of mathematics, the axiomatic approach having been taken for granted since the time of Euclid as the natural basis for mathematics. Core concepts such as axiom, order, and set received fresh emphasis. In mathematics as in physics, new and unexpected ideas had arisen and significant changes were coming. Inquiries into the consistency of mathematical theories lead to the development of a new level of study, a reflective critique in which the theory under review "becomes itself the object of a mathematical study", what Hilbert called metamathematics or proof theory (Kleene, 55).

At the midpoint of the century, a new mathematical theory known as category theory arose as a new contender for the natural language of mathematical thinking (Mac Lane 1998). As the 20th century progressed, however, philosophical opinions diverged as to just how well-founded were the questions about foundations that were raised at its opening. Hilary Putnam (1967) summed up one common view of the situation in the last third of the century by saying:

When philosophy discovers something wrong with science, sometimes science has to be changed — Russell's paradox comes to mind, as does Berkeley's attack on the actual infinitesimal — but more often it is philosophy that has to be changed. I do not think that the difficulties that philosophy finds with classical mathematics today are genuine difficulties; and I think that the philosophical interpretations of mathematics that we are being offered on every hand are wrong, and that 'philosophical interpretation' is just what mathematics doesn't need. (Putnam 1967/1996, 169–170).

Philosophy of mathematics today proceeds along several different lines of inquiry, by philosophers of mathematics, logicians, and mathematicians, and there are many schools of thought on the subject. The schools are addressed separately in the next section, and their assumptions explained.

Philosophy of mathematics (25 May 2006 b)

Philosophy of mathematics is the branch of philosophy that studies the philosophical assumptions, foundations, and implications of mathematics. Recurrent themes include:

What topics are in the scope of mathematics?
What does it mean to refer to a mathematical object?
What is a proper mathematical proof?
What are the fundamental assumptions or axioms used in mathematics?
What is the relationship between mathematics and logic?
What is the reason that mathematics is useful in the sciences?
What is mathematical beauty and truth?

This article focuses on the Western philosophy of mathematics.

Context

Many thinkers down through the ages have contributed their ideas concerning the conduct of mathematical inquiry and the nature of mathematical objects and knowledge. Today, many philosophers of mathematics aim to give accounts of this form of inquiry and its products as they stand, while others emphasize a role for themselves that goes beyond simple interpretation to critical analysis.

The terms philosophy of mathematics and mathematical philosophy are often taken to be synonymous, but others distinguish between them. The latter may be used to mean at least three distinct things. One sense refers to a project of formalizing a philosophical subject matter, say, aesthetics, ethics, logic, metaphysics, or theology, in a purportedly more exact and rigorous form, as for example the labors of Scholastic theologians, or the systematic aims of Leibniz and Spinoza. Another sense refers to the working philosophy of an individual practitioner or a like-minded community of practicing mathematicians. Additionally, some understand the term mathematical philosophy to be an allusion to the approach taken by Bertrand Russell in his book Introduction to Mathematical Philosophy.

Historical overview

Philosophy of mathematics in the 20th century

A perennial issue in the philosophy of mathematics concerns the relationship between logic and mathematics at their joint foundations. While 20th century philosophers continued to ask the questions mentioned at the outset of this article, the philosophy of mathematics in the 20th century is characterized by a predominant interest in formal logic, set theory, and foundational issues.

At the start of the century, philosophers of mathematics were already beginning to divide into various schools of thought about all these questions, broadly distinguished by their pictures of mathematical epistemology and ontology. Three schools, formalism, intuitionism, and logicism, emerged at this time, partly in response to the increasingly widespread worry that mathematics as it stood, and analysis in particular, did not live up to the standards of certainty and rigor that had been taken for granted. Each school addressed the issues that came to the fore at that time, either attempting to resolve them or claiming that mathematics is not entitled to its status as our most trusted knowledge.

Surprising and counterintuitive developments in formal logic and set theory early in the 20th century led to new questions concerning what was traditionally called the foundations of mathematics. As the century unfolded, the initial focus of concern expanded to an open exploration of the fundamental axioms of mathematics, the axiomatic approach having been taken for granted since the time of Euclid as the natural basis for mathematics. Core concepts such as axiom, order, and set received fresh emphasis. In mathematics as in physics, new and unexpected ideas had arisen and significant changes were coming. Inquiries into the consistency of mathematical theories lead to the development of a new level of study, a reflective critique in which the theory under review "becomes itself the object of a mathematical study", what Hilbert called metamathematics or proof theory (Kleene, 55).

At the midpoint of the century, a new mathematical theory known as category theory arose as a new contender for the natural language of mathematical thinking (Mac Lane 1998). As the 20th century progressed, however, philosophical opinions diverged as to just how well-founded were the questions about foundations that were raised at its opening. Hilary Putnam (1967) summed up one common view of the situation in the last third of the century by saying:

When philosophy discovers something wrong with science, sometimes science has to be changed — Russell's paradox comes to mind, as does Berkeley's attack on the actual infinitesimal — but more often it is philosophy that has to be changed. I do not think that the difficulties that philosophy finds with classical mathematics today are genuine difficulties; and I think that the philosophical interpretations of mathematics that we are being offered on every hand are wrong, and that 'philosophical interpretation' is just what mathematics doesn't need. (Putnam 1967/1996, 169–170).

Philosophy of mathematics today proceeds along several different lines of inquiry, by philosophers of mathematics, logicians, and mathematicians, and there are many schools of thought on the subject. The schools are addressed separately in the next section, and their assumptions explained.

Philosophy of mathematics (26 May 2006)

Philosophy of mathematics is the branch of philosophy that studies the philosophical assumptions, foundations, and implications of mathematics. Recurrent themes include:

What are the sources of mathematical subject matter?
What does it mean to refer to a mathematical object?
What is a proper mathematical proof?
What are the fundamental assumptions or axioms used in mathematics?
What is the relationship between mathematics and logic?
What is the reason that mathematics is useful in the sciences?
What is mathematical beauty and truth?

Context

Many thinkers down through the ages have contributed their ideas concerning the conduct of mathematical inquiry and the nature of mathematical objects and knowledge. Today, many philosophers of mathematics aim to give accounts of this form of inquiry and its products as they stand, while others emphasize a role for themselves that goes beyond simple interpretation to critical analysis.

The terms philosophy of mathematics and mathematical philosophy are often taken to be synonymous, but others distinguish between them. The latter may be used to mean at least three distinct things. One sense refers to a project of formalizing a philosophical subject matter, say, aesthetics, ethics, logic, metaphysics, or theology, in a purportedly more exact and rigorous form, as for example the labors of Scholastic theologians, or the systematic aims of Leibniz and Spinoza. Another sense refers to the working philosophy of an individual practitioner or a like-minded community of practicing mathematicians. Additionally, some understand the term mathematical philosophy to be an allusion to the approach taken by Bertrand Russell in his book Introduction to Mathematical Philosophy.

Perennial questions

A point of departure for the topical question of "Where mathematics comes from" can be taken from the following narrative, chosen for its typicality more than its novelty, of how abstractions are ostensibly extracted from the givens of raw experience.

Our concept of physical space is the result of a desire to order our experiences of the external world. This ordering process is accompanied by successive approximations and abstractions which lead to our concept of mathematical space. For the physicist the correspondence between the data of experience and his concept of physical space is all important. As the abstraction process continues, this correspondence becomes less significant, so that the mathematician feels free to concentrate upon the logical relations involved. (G. de B. Robinson, 5).

It is significant that the geometer speaks of a process of dual abstraction, both parallel and serial, that brings about a relationship among contingent experience, concepts of physical space, and concepts of mathematical space.

Historical overview

There are traditions of mathematical philosophy in both Western philosophy and Eastern philosophy. Western philosophizing about mathematics has a history that goes at least as far back as Plato, who considered the ontological status of mathematical objects, and Aristotle, who considered logic and issues related to infinity (actual versus potential). Greek views of quantity strongly influenced their views of other areas of mathematics. At one time, the Greeks held the opinion that 1 (one) was not a number, but rather a unit of arbitrary length (so that 3, for example, represented 3 such units and truly was a number). At another point, a similar argument was made that 2 was not a number but a fundamental notion of a pair. Of course, this was well before 0 was considered a number. These views come from the heavily geometric straight-edge-and-compass viewpoint of the Greeks: The first line drawn had unit length, and numbers represented multiples of it. Greek ideas of number were upended by the discovery of the irrationality of the square root of two, showing that the diagonal of a unit square was incommensurable with its (unit-length) edge: There was no number that represented how much longer the diagonal was than an edge. This caused a significant re-evaluation of Greek philosophy of mathematics, as non-Euclidean geometry would do to European philosophy of mathematics two millenia later.^{[citation needed]}

Beginning with Leibniz, the focus shifted strongly to the relationship between mathematics and logic. This view dominated the philosophy of mathematics through the time of Frege and of Russell, but was brought into question by developments in the late 19th and early 20th century.^{[citation needed]}

Philosophy of mathematics in the 20th century

A perennial issue in the philosophy of mathematics concerns the relationship between logic and mathematics at their joint foundations. While 20th century philosophers continued to ask the questions mentioned at the outset of this article, the philosophy of mathematics in the 20th century is characterized by a predominant interest in formal logic, set theory, and foundational issues.

At the start of the century, philosophers of mathematics were already beginning to divide into various schools of thought about all these questions, broadly distinguished by their pictures of mathematical epistemology and ontology. Three schools, formalism, intuitionism, and logicism, emerged at this time, partly in response to the increasingly widespread worry that mathematics as it stood, and analysis in particular, did not live up to the standards of certainty and rigor that had been taken for granted. Each school addressed the issues that came to the fore at that time, either attempting to resolve them or claiming that mathematics is not entitled to its status as our most trusted knowledge.

Surprising and counterintuitive developments in formal logic and set theory early in the 20th century led to new questions concerning what was traditionally called the foundations of mathematics. As the century unfolded, the initial focus of concern expanded to an open exploration of the fundamental axioms of mathematics, the axiomatic approach having been taken for granted since the time of Euclid as the natural basis for mathematics. Core concepts such as axiom, order, and set received fresh emphasis. In mathematics as in physics, new and unexpected ideas had arisen and significant changes were coming. Inquiries into the consistency of mathematical theories lead to the development of a new level of study, a reflective critique in which the theory under review "becomes itself the object of a mathematical study", what Hilbert called metamathematics or proof theory (Kleene, 55).

At the midpoint of the century, a new mathematical theory known as category theory arose as a new contender for the natural language of mathematical thinking (Mac Lane 1998). As the 20th century progressed, however, philosophical opinions diverged as to just how well-founded were the questions about foundations that were raised at its opening. Hilary Putnam (1967) summed up one common view of the situation in the last third of the century by saying:

When philosophy discovers something wrong with science, sometimes science has to be changed — Russell's paradox comes to mind, as does Berkeley's attack on the actual infinitesimal — but more often it is philosophy that has to be changed. I do not think that the difficulties that philosophy finds with classical mathematics today are genuine difficulties; and I think that the philosophical interpretations of mathematics that we are being offered on every hand are wrong, and that 'philosophical interpretation' is just what mathematics doesn't need. (Putnam 1967/1996, 169–170).

Philosophy of mathematics today proceeds along several different lines of inquiry, by philosophers of mathematics, logicians, and mathematicians, and there are many schools of thought on the subject. The schools are addressed separately in the next section, and their assumptions explained.

Philosophy of mathematics (29 May 2006)

Philosophy of mathematics is the branch of philosophy that studies the philosophical assumptions, foundations, and implications of mathematics. Recurrent themes include:

What are the sources of mathematical subject matter?
What does it mean to refer to a mathematical object?
What constitutes a valid mathematical proof?
What is the character of a mathematical proposition?
What are the fundamental assumptions or axioms used in mathematics?
What is the relationship between mathematics and logic?
What is the reason that mathematics is useful in the sciences?
What is mathematical beauty and truth?

Context

Many thinkers down through the ages have contributed their ideas concerning the conduct of mathematical inquiry and the nature of mathematical objects and knowledge. Today, many philosophers of mathematics aim to give accounts of this form of inquiry and its products as they stand, while others emphasize a role for themselves that goes beyond simple interpretation to critical analysis.

The terms philosophy of mathematics and mathematical philosophy are often taken to be synonymous, but others distinguish between them. The latter may be used to mean at least three distinct things. One sense refers to a project of formalizing a philosophical subject matter, say, aesthetics, ethics, logic, metaphysics, or theology, in a purportedly more exact and rigorous form, as for example the labors of Scholastic theologians, or the systematic aims of Leibniz and Spinoza. Another sense refers to the working philosophy of an individual practitioner or a like-minded community of practicing mathematicians. Additionally, some understand the term mathematical philosophy to be an allusion to the approach taken by Bertrand Russell in his book Introduction to Mathematical Philosophy.

Perennial questions

Three features of mathematical reasoning — its abstract, hypothetical, and necessary qualities — are so inseparable that their logical linkage is already a commonplace paradigm of Classical philosophy. The need to understand this complex of features leads to some of the initial encounters between mathematics and philosophy in general. For example, Plato bases one of his standard parables on the way that students of mathematics use visible forms as icons or images of formal realities:

The very things which they mould and draw, which have shadows and images of themselves in water, these things they treat in their turn as only images, but what they really seek is to get sight of those realities which can be seen only by the mind. (Plato, Republic 510E).

Plato is not engaged in the philosophy of mathematics, since mathematics is not his main object either here or elsewhere, and he is not proposing any brand of mathematical philosophy that aims to reduce a philosophical subject to mathematics. Plato's point is a wider one, having to do with the teaching that is now called Platonic realism in his honor. But he does find it useful to lift a theme out of Pythagoras' school, taking mathematics as a paradigmatic case of his broader philosophy.

A point of departure for the question of "Where mathematics comes from" can be taken from the following narrative, chosen for its typicality more than its novelty, of how abstractions are derived from the matrix of experience:

Our concept of physical space is the result of a desire to order our experiences of the external world. This ordering process is accompanied by successive approximations and abstractions which lead to our concept of mathematical space. For the physicist the correspondence between the data of experience and his concept of physical space is all important. As the abstraction process continues, this correspondence becomes less significant, so that the mathematician feels free to concentrate upon the logical relations involved. (G. de B. Robinson, 5).

The author describes a process of abstraction that produces empirically bound concepts and formally free concepts in tandem, and that brings about a threefold relation among contingent experiences, concepts of physical space, and concepts of mathematical space. Achieving a more thorough understanding of this process, by which mathematical patterns are abstracted from concrete experience, developed as quasi-autonomous forms, and then applied back to experience in far-reaching and surprising ways, is one of the essential services that philosophical examination can perform for the benefit of mathematical thought.

Mathematical propositions, at least at first sight, appear to differ from other sorts of propositions, but in ways that have, historically speaking, been difficult to define precisely. One distinctive feature of mathematical propositions is, as Hilary Putnam sketches a common view of it, "the very wide variety of equivalent formulations that they possess", by which he does not mean the sheer number of ways of saying the same thing but "rather that in mathematics the number of ways of expressing what is in some sense the same fact (if the proposition is true) while apparently not talking about the same objects is especially striking" (Putnam, 170).

Another characteristic of mathematical propositions, the recognition of which is drilled into the character of every student, is epitomized in the precept: "What's true is what you can prove". W.W. Tait (1986) takes up the relation between truth and proof in the process of examining the role of Platonism in mathematics.

Historical overview

Philosophy of mathematics in the 20th century

A perennial issue in the philosophy of mathematics concerns the relationship between logic and mathematics at their joint foundations. While 20th century philosophers continued to ask the questions mentioned at the outset of this article, the philosophy of mathematics in the 20th century is characterized by a predominant interest in formal logic, set theory, and foundational issues.

At the start of the century, philosophers of mathematics were already beginning to divide into various schools of thought about all these questions, broadly distinguished by their pictures of mathematical epistemology and ontology. Three schools, formalism, intuitionism, and logicism, emerged at this time, partly in response to the increasingly widespread worry that mathematics as it stood, and analysis in particular, did not live up to the standards of certainty and rigor that had been taken for granted. Each school addressed the issues that came to the fore at that time, either attempting to resolve them or claiming that mathematics is not entitled to its status as our most trusted knowledge.

Surprising and counterintuitive developments in formal logic and set theory early in the 20th century led to new questions concerning what was traditionally called the foundations of mathematics. As the century unfolded, the initial focus of concern expanded to an open exploration of the fundamental axioms of mathematics, the axiomatic approach having been taken for granted since the time of Euclid as the natural basis for mathematics. Core concepts such as axiom, order, and set received fresh emphasis. In mathematics as in physics, new and unexpected ideas had arisen and significant changes were coming. Inquiries into the consistency of mathematical theories lead to the development of a new level of study, a reflective critique in which the theory under review "becomes itself the object of a mathematical study", what Hilbert called metamathematics or proof theory (Kleene, 55).

At the midpoint of the century, a new mathematical theory known as category theory arose as a new contender for the natural language of mathematical thinking (Mac Lane 1998). As the 20th century progressed, however, philosophical opinions diverged as to just how well-founded were the questions about foundations that were raised at its opening. Hilary Putnam summed up one common view of the situation in the last third of the century by saying:

When philosophy discovers something wrong with science, sometimes science has to be changed — Russell's paradox comes to mind, as does Berkeley's attack on the actual infinitesimal — but more often it is philosophy that has to be changed. I do not think that the difficulties that philosophy finds with classical mathematics today are genuine difficulties; and I think that the philosophical interpretations of mathematics that we are being offered on every hand are wrong, and that 'philosophical interpretation' is just what mathematics doesn't need. (Putnam, 169–170).

Philosophy of mathematics today proceeds along several different lines of inquiry, by philosophers of mathematics, logicians, and mathematicians, and there are many schools of thought on the subject. The schools are addressed separately in the next section, and their assumptions explained.

Philosophy of mathematics (30 May 2006)

Philosophy of mathematics is the branch of philosophy that studies the philosophical assumptions, foundations, and implications of mathematics. Recurrent themes include:

What are the sources of mathematical subject matter?
What does it mean to refer to a mathematical object?
What is the character of a mathematical proposition?
What kinds of inquiry play a role in mathematics?
What are the fundamental assumptions or axioms used in mathematics?
What is the relationship between mathematics and logic?
What is the reason that mathematics is useful in the sciences?
What is mathematical beauty and truth?

Context

Many thinkers down through the ages have contributed their ideas concerning the conduct of mathematical inquiry and the nature of mathematical objects and knowledge. Today, many philosophers of mathematics aim to give accounts of this form of inquiry and its products as they stand, while others emphasize a role for themselves that goes beyond simple interpretation to critical analysis.

The terms philosophy of mathematics and mathematical philosophy are often taken to be synonymous, but others distinguish between them. The latter may be used to mean at least three distinct things. One sense refers to a project of formalizing a philosophical subject matter, say, aesthetics, ethics, logic, metaphysics, or theology, in a purportedly more exact and rigorous form, as for example the labors of Scholastic theologians, or the systematic aims of Leibniz and Spinoza. Another sense refers to the working philosophy of an individual practitioner or a like-minded community of practicing mathematicians. Additionally, some understand the term mathematical philosophy to be an allusion to the approach taken by Bertrand Russell in his book Introduction to Mathematical Philosophy.

Perennial questions

Three features of mathematical reasoning — its abstract, hypothetical, and necessary qualities — are so inseparable that their logical linkage is already a commonplace paradigm of Classical philosophy. The need to understand this complex of features leads to some of the initial encounters between mathematics and philosophy in general. For example, Plato bases one of his gnomic parables, the analogy of the divided line, on the way that students of mathematics use visible forms as images or simulacra of formal realities:

The very things which they mould and draw, which have shadows and images of themselves in water, these things they treat in their turn as only images, but what they really seek is to get sight of those realities which can be seen only by the mind. (Plato, Republic 510E).

Plato is not engaged in the philosophy of mathematics, since mathematics is not his main object either here or elsewhere, and he is not proposing the type of mathematical philosophy that aims to reduce philosophy in general to mathematics. Plato's point is a wider one, having to do with the teaching that is now called Platonic realism. But the form of analogy that maps reality into representation is a familiar theme in mathematics, and so it serves as the analogical image of a further analogy that Plato uses to illustrate his broader philosophy. Plato's reasoning in this part of the Republic is Plato at his subtlest, but it lays bare many of the founding metaphors of the Western tradition and will repay further consideration below.

This same form of argument, that Stanislaw Ulam (1990) would later dub analogies between analogies, brings our story right up to the present time frame, as mathematical category theory, a formalism that many mathematicians regard as the natural language of contemporary mathematics, is nothing more in the first instance than a formalization of mathematical metaphors.

Aristotle routinely derives his initial philosophical impulses from the parables of his predecessors, especially Plato, but his natural attraction to earthly topics just as dependably brings him back to empirical grounds. For a topical example, he starts from Plato's treatment of analogy as the mathematical form of a logos, a proportion, or a ratio but he goes on to analyze the pattern of reasoning by analogy or example — the Greek word he uses is παραδειγμα, the root of paradigms both grammatical and philosophical — as a mixed syllogism, in particular, a two-stage inference that follows a step of inductive reasoning with a step of deductive reasoning.

A point of departure for the question of "Where mathematics comes from" can be taken from the following narrative, chosen for its typicality more than its novelty, of how abstractions are derived from the matrix of experience:

Our concept of physical space is the result of a desire to order our experiences of the external world. This ordering process is accompanied by successive approximations and abstractions which lead to our concept of mathematical space. For the physicist the correspondence between the data of experience and his concept of physical space is all important. As the abstraction process continues, this correspondence becomes less significant, so that the mathematician feels free to concentrate upon the logical relations involved. (G. de B. Robinson, 5).

The author describes a process of abstraction that produces empirically bound concepts and formally free concepts in tandem, and that brings about a threefold relation among contingent experiences, concepts of physical space, and concepts of mathematical space. Achieving a more thorough understanding of this process, by which mathematical patterns are abstracted from concrete experience, developed as quasi-autonomous forms, and then applied back to experience in far-reaching and surprising ways, is one of the essential services that philosophical examination can perform for the benefit of mathematical thought.

Mathematical propositions, at least at first sight, appear to differ from other sorts of propositions, but in ways that have, historically speaking, been difficult to define precisely. One distinctive feature of mathematical propositions is, as Hilary Putnam sketches a common view of it, "the very wide variety of equivalent formulations that they possess", by which he does not mean the sheer number of ways of saying the same thing but "rather that in mathematics the number of ways of expressing what is in some sense the same fact (if the proposition is true) while apparently not talking about the same objects is especially striking" (Putnam, 170).

Another characteristic of mathematical propositions, the recognition of which is drilled into the character of every student, is epitomized in the precept: "What's true is what you can prove". W.W. Tait (1986) takes up the relation between truth and proof in the process of examining the role of Platonism in mathematics.

Placed within a broader context, proof may be seen as a form of inquiry, being any process that reduces the amount of uncertainty that a reasoner has about a given question. Viewing proof in this light leads to the further question: What other forms of inquiry are involved in the actual practice of mathematics?

Historical overview

Philosophy of mathematics in the 20th century

A perennial issue in the philosophy of mathematics concerns the relationship between logic and mathematics at their joint foundations. While 20th century philosophers continued to ask the questions mentioned at the outset of this article, the philosophy of mathematics in the 20th century is characterized by a predominant interest in formal logic, set theory, and foundational issues.

At the start of the century, philosophers of mathematics were already beginning to divide into various schools of thought about all these questions, broadly distinguished by their pictures of mathematical epistemology and ontology. Three schools, formalism, intuitionism, and logicism, emerged at this time, partly in response to the increasingly widespread worry that mathematics as it stood, and analysis in particular, did not live up to the standards of certainty and rigor that had been taken for granted. Each school addressed the issues that came to the fore at that time, either attempting to resolve them or claiming that mathematics is not entitled to its status as our most trusted knowledge.

Surprising and counterintuitive developments in formal logic and set theory early in the 20th century led to new questions concerning what was traditionally called the foundations of mathematics. As the century unfolded, the initial focus of concern expanded to an open exploration of the fundamental axioms of mathematics, the axiomatic approach having been taken for granted since the time of Euclid as the natural basis for mathematics. Core concepts such as axiom, order, and set received fresh emphasis. In mathematics as in physics, new and unexpected ideas had arisen and significant changes were coming. Inquiries into the consistency of mathematical theories lead to the development of a new level of study, a reflective critique in which the theory under review "becomes itself the object of a mathematical study", what Hilbert called metamathematics or proof theory (Kleene, 55).

At the midpoint of the century, a new mathematical theory known as category theory arose as a new contender for the natural language of mathematical thinking (Mac Lane 1998). As the 20th century progressed, however, philosophical opinions diverged as to just how well-founded were the questions about foundations that were raised at its opening. Hilary Putnam summed up one common view of the situation in the last third of the century by saying:

When philosophy discovers something wrong with science, sometimes science has to be changed — Russell's paradox comes to mind, as does Berkeley's attack on the actual infinitesimal — but more often it is philosophy that has to be changed. I do not think that the difficulties that philosophy finds with classical mathematics today are genuine difficulties; and I think that the philosophical interpretations of mathematics that we are being offered on every hand are wrong, and that 'philosophical interpretation' is just what mathematics doesn't need. (Putnam, 169–170).

Philosophy of mathematics today proceeds along several different lines of inquiry, by philosophers of mathematics, logicians, and mathematicians, and there are many schools of thought on the subject. The schools are addressed separately in the next section, and their assumptions explained.

Philosophy of mathematics (1 Jun 2006)

Philosophy of mathematics is the branch of philosophy that studies the philosophical assumptions, foundations, and implications of mathematics. Recurrent themes include:

What are the sources of mathematical subject matter?
What does it mean to refer to a mathematical object?
What is the character of a mathematical proposition?
What is the relation between logic and mathematics?
What kinds of inquiry play a role in mathematics?
What are the objectives of mathematical inquiry?
What gives mathematics its hold on experience?
What is the bearing of mathematical beauty?

Context

Many thinkers down through the ages have contributed their ideas concerning the conduct of mathematical inquiry and the nature of mathematical objects and knowledge. Today, many philosophers of mathematics aim to give accounts of this form of inquiry and its products as they stand, while others emphasize a role for themselves that goes beyond simple interpretation to critical analysis.

The terms philosophy of mathematics and mathematical philosophy are often taken to be synonymous, but others distinguish between them. The latter may be used to mean at least three distinct things. One sense refers to a project of formalizing a philosophical subject matter, say, aesthetics, ethics, logic, metaphysics, or theology, in a purportedly more exact and rigorous form, as for example the labors of Scholastic theologians, or the systematic aims of Leibniz and Spinoza. Another sense refers to the working philosophy of an individual practitioner or a like-minded community of practicing mathematicians. Additionally, some understand the term mathematical philosophy to be an allusion to the approach taken by Bertrand Russell in his book Introduction to Mathematical Philosophy.

Perennial questions

Three features of mathematical reasoning — its abstract, hypothetical, and necessary qualities — are so inseparable that their logical linkage is already a commonplace paradigm of Classical philosophy. The need to understand this complex of features leads to some of the initial encounters between mathematics and philosophy in general. For example, Plato bases one of his gnomic parables, the analogy of the divided line, on the way that students of mathematics use visible forms as images or simulacra of formal realities:

The very things which they mould and draw, which have shadows and images of themselves in water, these things they treat in their turn as only images, but what they really seek is to get sight of those realities which can be seen only by the mind. (Plato, Republic 510E).

Plato is not engaged in the philosophy of mathematics, since mathematics is not his main object either here or elsewhere, and he is not proposing the type of mathematical philosophy that aims to reduce philosophy in general to mathematics. Plato's point is a wider one, having to do with the teaching that is now called Platonic realism. But the form of analogy that maps reality into representation is a familiar theme in mathematics, and so it serves as the analogical image of a further analogy that Plato uses to illustrate his broader philosophy. Plato's reasoning in this part of the Republic is Plato at his subtlest, but it lays bare many of the founding metaphors of the Western tradition and will repay further consideration below.

This same form of argument, that Stanislaw Ulam (1990) would later dub analogies between analogies, brings our story right up to the present time frame, as mathematical category theory, a formalism that many mathematicians regard as the natural language of contemporary mathematics, is nothing more in the first instance than a formalization of mathematical metaphors.

Aristotle routinely derives his initial philosophical impulses from the parables of his predecessors, especially Plato, but his natural attraction to earthly topics just as dependably brings him back to empirical grounds. For a topical example, he starts from Plato's treatment of analogy as the mathematical form of a logos, a proportion, or a ratio but he goes on to analyze the pattern of reasoning by analogy or example — the Greek word he uses is παραδειγμα, the root of paradigms both grammatical and philosophical — as a mixed syllogism, in particular, a two-stage inference that follows a step of inductive reasoning with a step of deductive reasoning.

A point of departure for the question of "Where mathematics comes from" can be taken from the following narrative, chosen for its typicality more than its novelty, of how abstractions are derived from the matrix of experience:

Our concept of physical space is the result of a desire to order our experiences of the external world. This ordering process is accompanied by successive approximations and abstractions which lead to our concept of mathematical space. For the physicist the correspondence between the data of experience and his concept of physical space is all important. As the abstraction process continues, this correspondence becomes less significant, so that the mathematician feels free to concentrate upon the logical relations involved. (G. de B. Robinson, 5).

The author describes a process of abstraction that produces empirically bound concepts and formally free concepts in tandem, and that brings about a threefold relation among contingent experiences, concepts of physical space, and concepts of mathematical space. Achieving a more thorough understanding of this process, by which mathematical patterns are abstracted from concrete experience, developed as quasi-autonomous forms, and then applied back to experience in far-reaching and surprising ways, is one of the essential services that philosophical examination can perform for the benefit of mathematical thought.

Mathematical propositions, at least at first sight, appear to differ from other sorts of propositions, but in ways that have, historically speaking, been difficult to define precisely. One distinctive feature of mathematical propositions is, as Hilary Putnam sketches a common view of it, "the very wide variety of equivalent formulations that they possess", by which he does not mean the sheer number of ways of saying the same thing but "rather that in mathematics the number of ways of expressing what is in some sense the same fact (if the proposition is true) while apparently not talking about the same objects is especially striking" (Putnam, 170).

Another characteristic of mathematical propositions, the recognition of which is drilled into the character of every student, is epitomized in the precept: "What's true is what you can prove". W.W. Tait (1986) takes up the relation between truth and proof in the process of examining the role of Platonism in mathematics.

Placed within a broader context, proof may be seen as a form of inquiry, being one of many proceedings that reduce the amount of uncertainty a reasoner has about a given question. Viewing proof in this light leads to the further question: What other forms of inquiry are involved in the actual practice of mathematics? In particular, what are the roles of analogy, beauty, conjecture, and various types of experiential reasoning, from empirical induction to chance inspiration to concrete intuition, in the actual life of mathematical inquiry?

Philosophical inquiry into the grounds of mathematics sooner or later comes to a question about its relation to logic. The answers that suggest themselves naturally depend on the definitions of logic and mathematics that are in force at the time, or the basic intuitions about them if real definitions are yet to be found.

Historical overview

Philosophy of mathematics in the 20th century

A perennial issue in the philosophy of mathematics concerns the relationship between logic and mathematics at their joint foundations. While 20th century philosophers continued to ask the questions mentioned at the outset of this article, the philosophy of mathematics in the 20th century is characterized by a predominant interest in formal logic, set theory, and foundational issues.

At the start of the century, philosophers of mathematics were already beginning to divide into various schools of thought about all these questions, broadly distinguished by their pictures of mathematical epistemology and ontology. Three schools, formalism, intuitionism, and logicism, emerged at this time, partly in response to the increasingly widespread worry that mathematics as it stood, and analysis in particular, did not live up to the standards of certainty and rigor that had been taken for granted. Each school addressed the issues that came to the fore at that time, either attempting to resolve them or claiming that mathematics is not entitled to its status as our most trusted knowledge.

Surprising and counterintuitive developments in formal logic and set theory early in the 20th century led to new questions concerning what was traditionally called the foundations of mathematics. As the century unfolded, the initial focus of concern expanded to an open exploration of the fundamental axioms of mathematics, the axiomatic approach having been taken for granted since the time of Euclid as the natural basis for mathematics. Core concepts such as axiom, order, and set received fresh emphasis. In mathematics as in physics, new and unexpected ideas had arisen and significant changes were coming. Inquiries into the consistency of mathematical theories lead to the development of a new level of study, a reflective critique in which the theory under review "becomes itself the object of a mathematical study", what Hilbert called metamathematics or proof theory (Kleene, 55).

At the midpoint of the century, a new mathematical theory known as category theory arose as a new contender for the natural language of mathematical thinking (Mac Lane 1998). As the 20th century progressed, however, philosophical opinions diverged as to just how well-founded were the questions about foundations that were raised at its opening. Hilary Putnam summed up one common view of the situation in the last third of the century by saying:

When philosophy discovers something wrong with science, sometimes science has to be changed — Russell's paradox comes to mind, as does Berkeley's attack on the actual infinitesimal — but more often it is philosophy that has to be changed. I do not think that the difficulties that philosophy finds with classical mathematics today are genuine difficulties; and I think that the philosophical interpretations of mathematics that we are being offered on every hand are wrong, and that 'philosophical interpretation' is just what mathematics doesn't need. (Putnam, 169–170).

Philosophy of mathematics today proceeds along several different lines of inquiry, by philosophers of mathematics, logicians, and mathematicians, and there are many schools of thought on the subject. The schools are addressed separately in the next section, and their assumptions explained.

Philosophy of mathematics (5 Jun 2006)

Philosophy of mathematics is the branch of philosophy that studies the philosophical assumptions, foundations, and implications of mathematics. Recurrent themes include:

What are the sources of mathematical subject matter?
What does it mean to refer to a mathematical object?
What is the character of a mathematical proposition?
What kinds of inquiry play a role in mathematics?
What is the relationship between mathematics and logic?
What is the reason that mathematics is useful in the sciences?
What are mathematical beauty and truth?

Context

Many thinkers down through the ages have contributed their ideas concerning the conduct of mathematical inquiry and the nature of mathematical objects and knowledge. Today, many philosophers of mathematics aim to give accounts of this form of inquiry and its products as they stand, while others emphasize a role for themselves that goes beyond simple interpretation to critical analysis.

The terms philosophy of mathematics and mathematical philosophy are often taken to be synonymous, but others distinguish between them. The latter may be used to mean at least three distinct things. One sense refers to a project of formalizing a philosophical subject matter, say, aesthetics, ethics, logic, metaphysics, or theology, in a purportedly more exact and rigorous form, as for example the labors of Scholastic theologians, or the systematic aims of Leibniz and Spinoza. Another sense refers to the working philosophy of an individual practitioner or a like-minded community of practicing mathematicians. Additionally, some understand the term mathematical philosophy to be an allusion to the approach taken by Bertrand Russell in his book Introduction to Mathematical Philosophy.

Perennial questions

Three features of mathematical reasoning — its abstract, hypothetical, and necessary qualities — are so inseparable that their logical linkage is already a commonplace paradigm of Classical philosophy. The need to understand this complex of features leads to some of the initial encounters between mathematics and philosophy in general. For example, Plato bases one of his gnomic parables, the analogy of the divided line, on the way that students of mathematics use visible forms as images or simulacra of formal realities:

The very things which they mould and draw, which have shadows and images of themselves in water, these things they treat in their turn as only images, but what they really seek is to get sight of those realities which can be seen only by the mind. (Plato, Republic 510E).

Plato is not engaged in the philosophy of mathematics, since mathematics is not his main object either here or elsewhere, and he is not proposing the type of mathematical philosophy that aims to reduce philosophy in general to mathematics. Plato's point is a wider one, having to do with the teaching that is now called Platonic realism. But the form of analogy that maps reality into representation is a familiar theme in mathematics, and so it serves as the analogical image of a further analogy that Plato uses to illustrate his broader philosophy. Plato's reasoning in this part of the Republic is Plato at his subtlest, but it lays bare many of the founding metaphors of the Western tradition and will repay further consideration below.

This same form of argument, that Stanislaw Ulam (1990) would later dub analogies between analogies, brings our story right up to the present time frame, as mathematical category theory, a formalism that many mathematicians regard as the natural language of contemporary mathematics, is nothing more in the first instance than a formalization of mathematical metaphors.

Aristotle routinely derives his initial philosophical impulses from the parables of his predecessors, especially Plato, but his natural attraction to earthly topics just as dependably brings him back to empirical grounds. For a topical example, he starts from Plato's treatment of analogy as the mathematical form of a logos, a proportion, or a ratio but he goes on to analyze the pattern of reasoning by analogy or example — the Greek word he uses is παραδειγμα, the root of paradigms both grammatical and philosophical — as a mixed syllogism, in particular, a two-stage inference that follows a step of inductive reasoning with a step of deductive reasoning.

A point of departure for the question of "Where mathematics comes from" can be taken from the following narrative, chosen for its typicality more than its novelty, of how abstractions are derived from the matrix of experience:

Our concept of physical space is the result of a desire to order our experiences of the external world. This ordering process is accompanied by successive approximations and abstractions which lead to our concept of mathematical space. For the physicist the correspondence between the data of experience and his concept of physical space is all important. As the abstraction process continues, this correspondence becomes less significant, so that the mathematician feels free to concentrate upon the logical relations involved. (G. de B. Robinson, 5).

The author describes a process of abstraction that produces empirically bound concepts and formally free concepts in tandem, and that brings about a threefold relation among contingent experiences, concepts of physical space, and concepts of mathematical space. Achieving a more thorough understanding of this process, by which mathematical patterns are abstracted from concrete experience, developed as quasi-autonomous forms, and then applied back to experience in far-reaching and surprising ways, is one of the essential services that philosophical examination can perform for the benefit of mathematical thought.

Mathematical propositions, at least at first sight, appear to differ from other sorts of propositions, but in ways that have, historically speaking, been difficult to define precisely. One distinctive feature of mathematical propositions is, as Hilary Putnam sketches a common view of it, "the very wide variety of equivalent formulations that they possess", by which he does not mean the sheer number of ways of saying the same thing but "rather that in mathematics the number of ways of expressing what is in some sense the same fact (if the proposition is true) while apparently not talking about the same objects is especially striking" (Putnam, 170).

Another characteristic of mathematical propositions, the recognition of which is drilled into the character of every student, is epitomized in the precept: "What's true is what you can prove". W.W. Tait (1986) takes up the relation between truth and proof in the process of examining the role of Platonism in mathematics.

Placed within a broader context, proof may be seen as a form of inquiry, being one of many proceedings that reduce the amount of uncertainty a reasoner has about a given question. Viewing proof in this light leads to the further question: What other forms of inquiry are involved in the actual practice of mathematics? In particular, what are the roles of analogy, beauty, conjecture, and various types of experiential reasoning, from empirical induction to chance inspiration to concrete intuition, in the actual life of mathematical inquiry?

Philosophical inquiry into the grounds of mathematics sooner or later comes to a question about its relation to logic. The answers that suggest themselves naturally depend on the definitions of logic and mathematics that are in force at the time, or the basic intuitions about them if real definitions are yet to be found.

Historical overview

There are traditions of mathematical philosophy in both Western philosophy and Eastern philosophy. Western philosophizing about mathematics has a history that goes at least as far back as Plato, who considered the ontological status of mathematical objects, and Aristotle, who considered logic and issues related to infinity (actual versus potential). Greek views of quantity strongly influenced their views of other areas of mathematics. At one time, the Greeks held the opinion that 1 (one) was not a number, but rather a unit of arbitrary length (so that 3, for example, represented 3 such units and truly was a number). At another point, a similar argument was made that 2 was not a number but a fundamental notion of a pair. Of course, this was well before 0 was considered a number. These views come from the heavily geometric straight-edge-and-compass viewpoint of the Greeks: The first line drawn had unit length, and numbers represented multiples of it. Greek ideas of number were upended by the discovery of the irrationality of the square root of two, showing that the diagonal of a unit square was incommensurable with its (unit-length) edge: There was no number that represented how much longer the diagonal was than an edge. This caused a significant re-evaluation of Greek philosophy of mathematics, as non-Euclidean geometry would do to European philosophy of mathematics two millenia later.^{[citation needed]}

Beginning with Leibniz, the focus shifted strongly to the relationship between mathematics and logic. This view dominated the philosophy of mathematics through the time of Frege and of Russell, but was brought into question by developments in the late 19th and early 20th century.^{[citation needed]}

Philosophy of mathematics in the 20th century

A perennial issue in the philosophy of mathematics concerns the relationship between logic and mathematics at their joint foundations. While 20th century philosophers continued to ask the questions mentioned at the outset of this article, the philosophy of mathematics in the 20th century is characterized by a predominant interest in formal logic, set theory, and foundational issues.

At the start of the century, philosophers of mathematics were already beginning to divide into various schools of thought about all these questions, broadly distinguished by their pictures of mathematical epistemology and ontology. Three schools, formalism, intuitionism, and logicism, emerged at this time, partly in response to the increasingly widespread worry that mathematics as it stood, and analysis in particular, did not live up to the standards of certainty and rigor that had been taken for granted. Each school addressed the issues that came to the fore at that time, either attempting to resolve them or claiming that mathematics is not entitled to its status as our most trusted knowledge.

Surprising and counterintuitive developments in formal logic and set theory early in the 20th century led to new questions concerning what was traditionally called the foundations of mathematics. As the century unfolded, the initial focus of concern expanded to an open exploration of the fundamental axioms of mathematics, the axiomatic approach having been taken for granted since the time of Euclid as the natural basis for mathematics. Core concepts such as axiom, order, and set received fresh emphasis. In mathematics as in physics, new and unexpected ideas had arisen and significant changes were coming. Inquiries into the consistency of mathematical theories lead to the development of a new level of study, a reflective critique in which the theory under review "becomes itself the object of a mathematical study", what Hilbert called metamathematics or proof theory (Kleene, 55).

At the midpoint of the century, a new mathematical theory known as category theory arose as a new contender for the natural language of mathematical thinking (Mac Lane 1998). As the 20th century progressed, however, philosophical opinions diverged as to just how well-founded were the questions about foundations that were raised at its opening. Hilary Putnam summed up one common view of the situation in the last third of the century by saying:

When philosophy discovers something wrong with science, sometimes science has to be changed — Russell's paradox comes to mind, as does Berkeley's attack on the actual infinitesimal — but more often it is philosophy that has to be changed. I do not think that the difficulties that philosophy finds with classical mathematics today are genuine difficulties; and I think that the philosophical interpretations of mathematics that we are being offered on every hand are wrong, and that 'philosophical interpretation' is just what mathematics doesn't need. (Putnam, 169–170).

Philosophy of mathematics today proceeds along several different lines of inquiry, by philosophers of mathematics, logicians, and mathematicians, and there are many schools of thought on the subject. The schools are addressed separately in the next section, and their assumptions explained.

Contemporary schools of thought

Mathematical realism

Mathematical realism, like realism in general, holds that mathematical entities exist independently of the human mind. Thus humans do not invent mathematics, but rather discover it, and any other intelligent beings in the universe would presumably do the same. In this point of view, there is really one one sort of mathematics that can be discovered: Triangles, for example, are real entities, not the creations of the human mind.

Many working mathematicians have been mathematical realists; they see themselves as discoverers of naturally occurring objects. Examples include Paul Erdős and Kurt Gödel. Gödel believed in an objective mathematical reality that could be perceived in a manner analogous to sense perception. Certain principles (e.g., for any two objects, there is a collection of objects consisting of precisely those two objects) could be directly seen to be true, but some conjectures, like the continuum hypothesis, might prove undecidable just on the basis of such principles. Gödel suggested that quasi-empirical methodology could be used to provide sufficient evidence to be able to reasonably assume such a conjecture.

Within realism, there are distinctions depending on what sort of existence one takes mathematical entities to have, and how we know about them.

Platonism

Platonism is the form of realism that suggests that mathematical entities are abstract, have no spatiotemporal or causal properties, and are eternal and unchanging. This is often claimed to be the naive view most people have of numbers. The term Platonism is used because such a view is seen to parallel Plato's belief in a "World of Ideas", an unchanging ultimate reality that the everyday world can only imperfectly approximate. Plato's view probably derives from Pythagoras, and his followers the Pythagoreans, who believed that the world was, quite literally, built up by the numbers.

It should be noted that this reading of "Platonism" is rejected by modern philosopher Alain Badiou, who considers the "empiricist" relationship between object and subject (where objects external to one's mind act, through the senses, on an internal subjective realm) utterly foreign to Platonic thought, according to which this location of mathematical entities is irrelevant to their ontological status. Badiou, in fact, identifies mathematics with ontology, considering mathematical discovery to be the scientific investigation of Being qua Being.

The major problem of mathematical platonism is this: precisely where and how do the mathematical entities exist, and how do we know about them? Is there a world, completely separate from our physical one, which is occupied by the mathematical entities? How can we gain access to this separate world and discover truths about the entities?

Gödel's platonism postulates a special kind of mathematical intuition that lets us perceive mathematical objects directly. (This view bears resemblances to many things Husserl said about mathematics, and supports Kant's idea that mathematics is synthetic a priori.) Davis and Hersh have suggested in their book The Mathematical Experience that most mathematicians act as though they are Platonists, even though, if pressed to defend the position carefully, they may retreat to formalism (see below).

Some mathematicians hold opinions that amount to more nuanced versions of Platonism. These ideas are sometimes described as Neo-Platonism.

Logicism

One of the most important questions for the foundations of mathematics is that of the relation between mathematics and logic. Logicism is the thesis that mathematics is reducible to logic, hence nothing but a part of logic. (Carnap 1931/1883, 41).

Rudolf Carnap (1931) presents the logicist thesis in two parts:

1.	The concepts of mathematics can be derived from logical concepts through explicit definitions.
2.	The theorems of mathematics can be derived from logical axioms through purely logical deduction.

If mathematics is a part of logic, then questions about the reality of mathematical objects reduce to questions about the reality of logical objects. But what, if anything, are the objects of logical concepts?