He never used the word "approximately", and I think you are reading the statement(s) incorrectly because of that. Let me be more specific.

The article says:

    The fundamental theorem of arithmetic
    states that every positive integer can
    be factorized in one way as a product
    of prime numbers.

OK, so that is an initial statement to give the general idea of what's going on. There are some details that need now to be added to avoid misunderstanding. In particular, the statement needs to be interpreted carefully. Gowers goes on to say:

    This statement has to be appropriately
    interpreted: ...

That's what I just said above ...

    ... we count the factorizations 3x5x13
    and 13x3x5 as the same, for instance.

So this is an example to set the scene for what is to follow. It points out that in the statement of the FTA we are going to regard to factorisations that differ only in the ordering of the factors as being the same.

Reading your other comments, it appears you are arguing against something that you actually agree with, and I suspect you have mis-read or mis-interpreted what Gowers has written.

It doesn't say that. What it says is:

    The fundamental theorem of arithmetic states that
    every positive integer can be factorized in one
    way as a product of prime numbers. This statement
    has to be appropriately interpreted: we count the
    factorizations 3x5x13 and 13x3x5 as the same, for
    instance.

That's not the same thing at all.

I miss quoted the original blog by replacing "has" for "can" and such. But the second sentence of what you quoted is misleading because that's not an correct interpretation of FTA.

    > But the second sentence of what you
    > quoted is misleading because that's
    > not an correct interpretation of FTA.

I wonder if this is a language thing. The second sentence says:

    ... we count the factorizations 3x5x13
    and 13x3x5 as the same, for instance.

This is giving an simple example of what the phrase "up to ordering" implies. The full statement of the FTA says that the factorisation is unique "up to ordering" and that's exactly what the second sentence is saying.

So I still think you are mistaken, and I don't understand why you are disagreeing with what Gowers wrote. Perhaps you could give more detail about why you think he is wrong.

You also misquoted "appropriately" as "approximately", making the meaning completely different.

Where did you read that?

https://www.google.com/url?sa=t&source=web&rct=j&url=http://... note that prime factorization is unique up to order. We aren't talking about order when we use the term uniqueness.

... which is exactly what Gowers is saying in the second sentence.