QuestionsCan we be certain of anything?A perception of mathematical objects: Gödel's intuition
Can we be certain of anything?Yes, through reason

A perception of mathematical objects: Gödel's intuition

Mathematical intuition; Gödel's Platonism

Kurt Gödel 1906 · 1978A logician who gave a formal version of the ontological proof.
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What it claims

Some mathematical truths are known with certainty, by reason, through an intuition that is like a perception of mathematical objects.

Why

The axioms of set theory force themselves upon us as true, like a perception: there is no reason to trust it less than the senses.

The same idea through history
Plato380 BCThe idea that mathematical objects exist outside us and that the mind can grasp them goes back to Plato. Gödel gives a modern version of it, tied to set theory.
Kurt Gödel1964Gödel adds the famous passage on the 'perception' of sets in the 1964 revised version of his paper on Cantor's continuum problem.
In short

"A perception of mathematical objects: Gödel's intuition" is an argument by Kurt Gödel ('What is Cantor's Continuum Problem?' (1947, revised version 1964), 1964) on the question "Can we be certain of anything?" (answer: yes, through reason). Its claim: Some mathematical truths are known with certainty, by reason, through an intuition that is like a perception of mathematical objects. According to AskPhilosophy's logic check (Z3 solver), the argument is valid only if one grants a premise the author leaves unstated (P4). AskPhilosophy presents 3 objections, 1 of them with a documented reply, and the original text behind each premise, checked word for word.

'What is Cantor's Continuum Problem?' (1947, revised version 1964)
Valid with an implicit premise

How do we know that 2 + 2 = 4, or that one infinite set is larger than another? We neither see nor touch numbers. Gödel, one of the greatest logicians of the twentieth century, holds that we nevertheless have a sort of perception of mathematical objects: when we reflect on the notions of set or number, certain axioms force themselves upon us as true, as a table imposes itself on sight. This 'mathematical intuition' is no less trustworthy than sense perception, which leads us to build physical theories. Reason can therefore reach objective mathematical truths. Gödel does not call it infallible: like perception it corrects itself, and a new axiom can also be justified by its fruitful consequences.

P1Despite their remoteness from sense experience, we have 'something like a perception' of the objects of set theory.
P2This is seen from the fact that the axioms force themselves upon us as being true.
P3There is no reason to have less confidence in this perception, mathematical intuition, than in sense perception, which induces us to build up physical theories.
P4 · implicit premiseWhat forces itself upon us as true through a perception as reliable as the senses is known with certainty.
Therefore some mathematical truths are known with certainty, by reason, through an intuition that is like a perception of mathematical objects.

click a premise to reject it

If you accept every premise, logic forces you to accept the conclusion. So the debate is about the premises.
Checked by a logic solver: the premises cannot all be true while the conclusion is false.
Valid with P4. P1 to P3 set up an analogy between mathematical intuition and perception. P4, hidden, turns this analogy into certainty. This is the weak point: sense perception sometimes deceives us (illusions), and so do mathematical intuitions (Russell's paradox grew out of an axiom that seemed evident). The argument thus mainly establishes reasonable confidence; it is filed under 'certainty' because Gödel holds that these truths are known by reason, not by convention or by sense experience.
Analogy. A mathematician facing an axiom is like a hiker facing a landscape: he does not invent it, he discovers it. If he is wrong about a distant detail, that does not mean there is no landscape.

Objections

3

Each objection is a conversation: click to see who replies and where the debate stands.

P
Paul Benacerraf 1973 · attacks premise 1

In 'Mathematical Truth', Benacerraf raises the access problem: if mathematical objects exist outside space and time, they have no causal effect on us. How could we perceive them? Perception requires a causal relation with its object; Gödel's intuition does not say which.

no reply
We have not yet found a direct reply by Gödel in the texts.
The debate goes on (our summary) Gödel offers no causal explanation in this passage. A defender would add (our reconstruction) that the perception of mathematical objects is no stranger than that of physical objects, whose perception already presupposes concepts that the senses do not supply (our reading). The problem raised by Benacerraf remains, for many philosophers, without an accepted answer.
Where the debate stands (our summary)Benacerraf: without a causal link, no perception. Gödel: intuition is no less reliable than the senses. Core disagreement: does knowledge require a causal link with its object?
Bertrand Russell 1902 · attacks premise 4

Russell's paradox shows that an axiom that forces itself upon us as evident (every predicate defines a set) can be contradictory. If intuitions about sets force themselves on us but lead to contradiction, they do not guarantee truth.

a defender replies
Kurt Gödel defender of the argument · 1964
Gödel answers in the same text that the set-theoretical paradoxes are hardly any more troublesome for mathematics than deceptions of the senses are for physics. The paradox struck a naive notion of set; the iterative notion (sets built in stages) leads to no known contradiction. Critics answer that this admits intuition does not give certainty at once.
Where the debate stands (our summary)Russell: an evident intuition can lead to contradiction. Gödel: it is corrected by a better intuition. Core disagreement: can a source that errs ground certainty?
P
Penelope Maddy 1990 · attacks premise 3

Maddy defended in Realism in Mathematics (1990) a naturalised version of intuition: we perceive sets of physical objects (three eggs in a carton), and intuition of sets is explained by the brain, not by a special faculty. In Naturalism in Mathematics (1997) she gives up this realism: the aims and practice of mathematicians, not an intuition, justify the axioms.

no reply
No reply possible: objection raised in 1990, after the author's death (1978).
Where the debate stands (our summary)Maddy: Gödel's intuition should be replaced by a natural basis, then by practice. Core disagreement: what justifies an axiom, a perception or its usefulness?