nLab epistemology of mathematics



The epistemology of mathematics is the study of mathematical knowledge.

There are a few branches of epistemology which could be applied to mathematics and the foundations of mathematics.


Rationalism states that reason is what determines what knowledge is. As formal mathematics entirely relies on deduction, such as natural deduction, to derive all of mathematics from some collection of basic primitives and axioms in the foundations of mathematics, it relies on reason to derive mathematical knowledge.


Empiricism states that experience is what determines what knowledge is.

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From the point of view of the history of mathematics, many mathematical concepts are first developed from physical intuition, and then only later are they formalized in logic. For example, Newton and Leibniz developed calculus on the derivative and the Riemann integral from the experience of geometry and physics in real life, and only in the 19th century was calculus (in particular the derivative and the Riemann integral) put on a formal rationalist basis with epsilontic analysis.

The pipeline from physics to pure mathematics abounds with examples which demonstrate empiricism. Similar things have occurred with Minkowski geometry and Riemannian geometry in the early 20th century with the advent of special relativity and general relativity, and with differential cohomology in the mid 20th century with the advent of quantum field theory.


Pragmatism states that all mathematical knowledge, including the words and language and symbols and notation associated with mathematics, are tools for mathematicians to use to solve problems in mathematics and to communicate to each other.

Clarke-Duane 2022 argued that pluralism implies Carnap’s pragmatism, as the relevant questions are not whether certain axioms are true or rules are derivable, but rather normative statements of which collection of axioms or rules to use in the foundations of mathematics in the broader mathematics community.

Epistemic relativism

Epistemic relativism states that what is true or justified for one person is not necessarily true or justified for another person.

Looking at the mathematical community as a whole, epistemic relativism seems to be true in formal mathematics and in the foundations of mathematics. Take excluded middle as an example. For classical mathematicians working in ZFC, excluded middle is simply true. However, for constructive mathematicians working in Brouwer’s intuitionism, or in the sheaf topos of the Dedekind real numbers, or in a smooth topos, excluded middle is provably false. In neutral constructive mathematics, excluded middle can neither be proven true nor false (it is independent), and neither it nor its negation is justified. Thus, the differing status of excluded middle for different mathematicians demonstrates epistemic relativism in mathematics.

Epistemic idealism

For now see idealism, absolute idealism, subjective idealism, objective idealism.


See also:

Last revised on December 28, 2022 at 21:32:24. See the history of this page for a list of all contributions to it.