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According to OED, noumenon is

An object knowable only by the mind or intellect, not by the senses

But I'm a little confused at considering about numbers, they seem to be objects knowable only by the mind or intellect; but when we see 2 apples on a table, we seem to can know 2 by our sense of sight. But I'm not sure whether the number 2 of 2 apples is truly known by our sense of sight.

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    for a great discussion see: philosophy.stackexchange.com/questions/1/…
    – Dr Sister
    Oct 10, 2012 at 2:17
  • I'm not even sure what this means given modern knowledge of neurobiology. We can't interpret sensory input without a mind. Nothing is knowable by the senses alone. If you say that a noumenon is something that can be known without any sensory input, then numbers are certainly included. Then again, try to generate a human mind without any sensory input...not likely to work very well.
    – Rex Kerr
    Oct 14, 2012 at 9:31
  • @RexKerr Your view, I think, make a sense, but I still have some doubts at whether numbers can be known without any sensory input. For example, we learnt 1,2,3,... by count sticks or coins in primary schools, but if we cognize them just directly by their definition instead of count sticks in the beginning period, can we really know them?
    – Popopo
    Oct 17, 2012 at 9:18
  • @Popopo - If you don't know numbers by their definition and properties, I'd say you don't really know numbers. If you need to fall back on sensory experience of counting sticks, you're missing the point--you don't actually know numbers, you just know sensory experience part of which can be nicely abstracted by using numbers.
    – Rex Kerr
    Oct 17, 2012 at 14:26
  • @RexKerr OK, but there is a question remains, that is, is it sufficient to know a number only by its definition? Know a definition, in my view, does not equivalent to know the meaning of the definition. e.g., we can tell a baby that 'zero is the empty set', but how can he/she know the meaning of the word 'empty'? Does he/she have to get help from concrete representations such as an empty box or an empty cup?
    – Popopo
    Oct 18, 2012 at 4:08

2 Answers 2

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The answer to this will depend on the metaphysical commitments you have. It is hard to see how a physicalist, or signatory to any form of metaphysical monism, i.e. to the belief that reality at root consists of one kind of stuff, can find room for numbers to be revealed to us through experience without being experienced themselves.

If everything is reducible to physics, then so is the number we count to when we see two apples; if this is physical in nature, then we experience this just as we experience their redness or acidic taste. Someone with this belief might further want to say that numbers are not things that exist apart from the physical entities that they bear on, rather that they supervene on top of these 'bona fide' physical things. The twoness of two apples is just a relation that obtains between these physical things, rather than something separately existing.

Just some thoughts to get started with.

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    I don't think your thoughts apply. A physicalist would not accept the idea of noumena in general, and so there is nothing peculiar about numbers to his rejection. I think an answer is only interesting (andthe question non-trivial) for metaphysics that permit noumena. I.e. I think more formally, the OPs question is: "do metaphysics that accept noumena necessarily have to classify numbers as noumena?"
    – Artem Kaznatcheev
    Oct 12, 2012 at 5:32
  • Nope. Numbers are 1) facts of knowledge (metaphysical) and 2) related to the phenomenon (thing as it is perceived), while noumenON is 1) a fact of the world (physical) and 2) related to the thing-as-it-is. We can't even know if the noumenon has a one-to-one relationship, that is, if the thing which can be numbered in the mind is enumerable in the noumenon (there are no numbers out there, perhaps everything is just quantum fields, no watermelons or lambs exist as such).
    – RodolfoAP
    Sep 29 at 23:45
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As Guambra Feo has pointed out, metaphysical presupposition play a role in determining an answer. For instance, the term 'noumenon' invokes the term 'real' or 'objective' indirectly by referencing 'sense-independence'. From WP:

In philosophy, a noumenon (/ˈnuːmənɒn/, /ˈnaʊ-/; from Ancient Greek νoούμενον; PL: noumena) is knowledge posited as an object that exists independently of human sense.

So, one's ontological commitments must be made manifest before an answer can be given. A mathematical realist would be in the position of answering yes. If numbers are somehow real as Platonic forms or otherwise, then there would be noumenal aspect to them, as they exist independently of human cognition and sense. For a mathematical fictionalist as in Hartry, numbers have no noumenal aspect as they are entirely phenomenological in the sense that they are a function of cognition and are not independent at all of cognition.

Thus, you wind up at the great obviator of philosophical discourse, "it depends".

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