Paolo Mancosu - The Philosophy of Mathematical Practice
Mathematical Explanation: Why it Matters
Introduction
● Two senses of mathematical explanation:
○ explanation in natural or social sciences which relies on mathematical
facts
○ explanation within mathematics
Mathematical explanations of scientific facts
● Some physical explanations are non causal
○ for example, if we throw a bunch of sticks into the air and freeze the
scene at any point, more of the sticks will be closer to the horizontal than the vertical.
Why? Because there are more ways for a stick to be horizontal than vertical. This is a
geometric fact, not a cause
● Articulating how such explanations work is difficult because it requires us to account for
how maths hooks onto reality
● Steiner claimed that an explanation of a physical fact is mathematical if when one strips
away the physics, one is left with a mathematical explanation of a mathematical fact
○ but through positing mathematical facts, does one imply the existence of
mathematical entities?
■ Steiner claims not, because to talk in terms of
explanation actually presupposes, rather than implies, the existence of
mathematical entities. Hence insofar as we explain we must accept the existence
of such entities
● Baker’s indispensability argument claims that ‘Mathematics is indispensable for our best
science. We ought to believe our best scientific theories and therefore we ought to accept the
kind of entities our best theories quantify over.’
○ this view is supported by the idea that postulating mathematical entities
might result in an increase of ‘theoretical virtues’ including explanatory power (Melia,
Colyvan)
● So the argument so far is:
○ there are genuinely mathematical explanations of empirical phenomena
○ we ought to be committed to the theoretical posits of these explanations
○ therefore we ought to be committed to the mathematical entities
postulated by these explanations
● Are we committed to the real existence, or the fictional existence, of the posits of such
explanations?
From mathematical explanations of scientific facts to mathematical explanations of mathematical facts
● Is the indispensability argument question begging?
○ we cannot use the existence of genuine explanations to imply
mathematical entities, because ‘genuine’ explanation implies true explanans
● (IF we assume the existence of some basic mathematical entities) we can argue via
explanatory power for the existence of other, more abstract entities, in a different indispensability
Mathematical Explanation: Why it Matters
Introduction
● Two senses of mathematical explanation:
○ explanation in natural or social sciences which relies on mathematical
facts
○ explanation within mathematics
Mathematical explanations of scientific facts
● Some physical explanations are non causal
○ for example, if we throw a bunch of sticks into the air and freeze the
scene at any point, more of the sticks will be closer to the horizontal than the vertical.
Why? Because there are more ways for a stick to be horizontal than vertical. This is a
geometric fact, not a cause
● Articulating how such explanations work is difficult because it requires us to account for
how maths hooks onto reality
● Steiner claimed that an explanation of a physical fact is mathematical if when one strips
away the physics, one is left with a mathematical explanation of a mathematical fact
○ but through positing mathematical facts, does one imply the existence of
mathematical entities?
■ Steiner claims not, because to talk in terms of
explanation actually presupposes, rather than implies, the existence of
mathematical entities. Hence insofar as we explain we must accept the existence
of such entities
● Baker’s indispensability argument claims that ‘Mathematics is indispensable for our best
science. We ought to believe our best scientific theories and therefore we ought to accept the
kind of entities our best theories quantify over.’
○ this view is supported by the idea that postulating mathematical entities
might result in an increase of ‘theoretical virtues’ including explanatory power (Melia,
Colyvan)
● So the argument so far is:
○ there are genuinely mathematical explanations of empirical phenomena
○ we ought to be committed to the theoretical posits of these explanations
○ therefore we ought to be committed to the mathematical entities
postulated by these explanations
● Are we committed to the real existence, or the fictional existence, of the posits of such
explanations?
From mathematical explanations of scientific facts to mathematical explanations of mathematical facts
● Is the indispensability argument question begging?
○ we cannot use the existence of genuine explanations to imply
mathematical entities, because ‘genuine’ explanation implies true explanans
● (IF we assume the existence of some basic mathematical entities) we can argue via
explanatory power for the existence of other, more abstract entities, in a different indispensability
