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The real numbers are constructed from the rational numbers which are constructed from the integers which, in turn, are constructed from the natural numbers. But if we had no notion of the natural numbers to begin with, could we "come up with" the real numbers?

I think maybe yes, in the way of just as sequences of the digits $0$ and $1$ (the binary system might be the easiest). These are just symbols, no connection to the natural numbers. But could we think the real numbers as a continuous number line in this way?

ploosu2
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    It's hard for me to imagine a universe where the natural numbers are not the first numbers known. – vadim123 Apr 21 '15 at 21:07
  • @vadim123: I find it impossible to imagine a universe in which the natural numbers could exist (other than as idealised mental abstractions). Where would you put them all? – Rob Arthan Apr 21 '15 at 21:11
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    @plousu2: you said it! You could "come up with" the real numbers by reflecting on the properties of an idealised straight line and segments thereof without needing to count things. – Rob Arthan Apr 21 '15 at 21:14
  • @RobArthan: This might be true, but I have a feeling that it's very hard to talk of almost any property of real numbers without using natural numbers. For instance, we can surely define order and sum of two real numbers, but product is already a tricky business ("the number associated with the area of a rectangle with the given lengths" is already quite a stretch). Now what about completeness? Well, easy, let's just check Cauchy sequences... oops, but sequences use natural numbers! – Pedro M. Apr 21 '15 at 21:45
  • @Pedro M: there is nothing in the OP's question about completeness. You might like to read up on tame geometry, – Rob Arthan Apr 21 '15 at 21:59
  • Excellent, difficult question. This led me to thinking that maybe we could be aware of rational numbers without being aware of integers, but that seems preposterous on its face. The integers are of such fundamental importance that Leopold Kronecker once said "die ganzen Zahlen hat der liebe Gott gemacht, alles andere ist Menschenwerk." – Lisa Apr 21 '15 at 22:21

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I think you could make the argument that not only is this possible, but it has been done, and 2300 years ago at that! I am referring to Euclidean geometry. We think of real numbers as measuring the lengths of line segments; or, to use more Euclidean language, telling us when to line segments are congruent. In the Elements, Euclid frequently refers indirectly to real numbers. For example, Proposition 3 of Book I is:

To cut off from the greater of two given unequal straight lines a straight line equal to the less.

You can think of this as explaining how to define subtraction. Given two line segments, Euclid shows us how find a line segment whose length is equal to the difference of the two given line segments' lengths.

You can also define multiplication and division of real numbers if you're allowed to designate a particular line segment as having unit length. You can do this with similar triangles; see this question for a nice picture.

Sure, you can define integers within Euclidean geometry as multiples of a particular length, but this is often not relevant. The Elements treats the concepts of similar triangles, area, etc., long before getting around to talking about integers.

You might object that Euclid was very nonrigorous by modern standards, but it's possible to formalize Euclidean geometry in a rigorous way that does not use the real numbers or natural numbers; see Hilbert's Axioms.

I think it's possible to imagine a society existing with a notion of length but without anyone thinking in terms of integers. For example, instead of counting out currency in distinct units like coins, your currency might be lengths of string of some type. You might argue that many animals have an intuitive sense of length without understanding integers, so in that sense I would say that real numbers are the more primitive notion.

If you want to actually construct the real numbers, say from the axioms of set theory, that is another matter. Maybe it could be done somehow without first creating the integers, but I'm not sure what that would accomplish. Euclid and Hilbert take the more congenial root of giving axioms for geometry without showing there's a model, meaning a mathematical object that satisfies the axioms.

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Jair Taylor
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  • Historically, currency really was a continuous quantity, specifically the weight of some precious metal such as gold. It turns out that commerce is significantly more convenient if you have standardized coins with standardized weights, and then you end up with integers in practice. But the continuous came before the discrete. – Kevin Oct 13 '24 at 23:58