I don’t know about you, I can work with it just fine. I know its properties. I can manipulate it. I can prove theorems about it. What more is there?

In fact, if you are to argue that we cannot know a “raw” real number, I would point out that we can’t know a natural number either! Take 2: you can picture two apples, you can imagine second place, you can visualize its decimal representation in Arabic numerals, you can tell me all its arithmetical properties, you can write down its construction as a set in ZFC set theory… but can you really know the number – not a representation of the number, not its properties, but the number itself? Of course not: mathematical objects are their properties and nothing more. It doesn’t even make sense to consider the idea of a “raw” object.

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Eddy_Viscosity2 7 days ago | parent | next [-]

You can hold a two in your head, but you can't hold a number with infinitely many decimal places. Any manipulations you do with the real 2 are done conceptually whereas with the natural 2, its done concretely.

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empath75 6 hours ago | parent | next [-]

You absolutely cannot visualize the concept "2" in your head. You can only visualize "two things" or the word "two" or the symbol "2", but the actual concept "2" is just as abstract as a real number.

You might say, I can imagine 2 apples, but I can't imagine pi apples, but you could just as easily imagine unrolling a circle with a diameter of 1, and you have visualized "pi" just as well as you can visualize 2 apples.

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drdeca 7 days ago | parent | prev [-]

The decimal places are just a way of representing it.

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Eddy_Viscosity2 6 days ago | parent [-]

The infinite number of decimal places is the definitional feature of a real number. No matter how's it represented they are still there and cannot be contained in our brains. We can say pi and hold the concept of pi in our heads, but not the actual number.

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drdeca 6 days ago | parent [-]

No, it really isn’t. The real numbers can be constructed in a number of ways, and it is more common to define them as either Dedekind cuts, or equivalence classes of Cauchy sequences of rational numbers.

Personally, I’d go with the sideline cut definition.

	▲	drdeca 6 days ago \| parent [-]
		Dang autocorrect. “sideline” should be Dedekind

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spyrja 6 days ago | parent | prev [-]

Or maybe we can know them equally well? The function f(x) = x(0^(sin(πx)^2)) for example "requires" infinities, but only returns integer values.