> If we'd chosen some other definition of multiplication, a lot of the "intuitive" properties of multiplication that hold over the natural numbers (such as the distributivity of multiplication over addition and subtraction) would no longer be true over the integers.
This is backward reasoning. The chosen definition of multiplication is not to keep things "intuitive". If you start with the field axioms, the chosen definition of multiplication is pretty much dictated by the axioms. If you choose another definition of multiplication, you would end with contradictions like 1 = 0 and such nonsense! And mathematicians abhor contradictions!
"Product of additive inverses of two elements is equal to the product of the two elements" is dictated by the field axioms in all fields.
Not every arithmetic property needs to be proved from Peano axioms. One can but it is tedious and unnecessary. A much better starting point is the set of field axioms where the distributivity property is already available as an axiom.
The assumptions made in the article are perfectly fine as per field axioms. Granted it would have been nicer if distributivity over addition was used instead of distributivity over subtraction. But it is not a big leap to derive distributivity over substraction from field axioms by distributing multiplication over a positive number and the additive inverse of another positive number.
Wherever you see an assumption made about negative number, just mentally replace it with additive inverse of a positive number and you would be fine.
1. You literally cannot prove this fact from the Peano axioms because Peano arithmetic operates on natural numbers, not integers.
2. As I said in my original post at the top, negative numbers are different from the unary subtraction operator (the additive inverse in the field). The number -2 is an entity that exists by itself regardless of whether you've defined an additive inverse. It turns out that the additive inverse of every positive integer is the corresponding negative integer, but this follows from the definition of +, not the other way around.
3. Even if you give OP the benefit of the doubt regarding his dodgy proof, it is saying something about the additive inverse and its relation to multiplication. It is not saying why the result of multiplying two negative values must be positive.
4. The multiplicative operator over the field must already be defined for you to be able to prove distributivity over addition. You can't assume distributivity over an operator that is only partially defined.
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Think about how you'd define a field. First, you need a set (let's call it Z), then you need two total operators over the set (+ and ), and two elements of the set (0 and 1) and each of these must satisfy specific properties (aka the field axioms). In particular, + and must be defined for all members of Z, not just Z+ and further + and * must be distributive. These are all facts you need to prove about Z, *, +, and 1 and only then do you have a field. You cannot work backward by assuming the field axioms (which are unfortunately named because they are not axioms at all but properties) to derive the definition o the field operators.
> You literally cannot prove this fact from the Peano axioms because Peano arithmetic operates on natural numbers, not integers.
Sure you can. With definitions! Define integers from natural numbers. Define rationals from integers. And so on. And so on.
> The number -2 is an entity that exists by itself regardless of whether you've defined an additive inverse.
I see a serious misunderstanding of this topic. Please read upon the field axioms and ring axioms if you haven't so already. Then please check https://math.stackexchange.com/a/878844 which is arguably more rigorous than this post. But the essence is the same. This is more rigorous because the subtraction operator is not used anywhere. Only addition, multiplication and additive inverses have been used. Like another commenter said, if you just replace subtraction with addition with an additive inverse in the OP's post, things fall in place.
> You cannot work backward by assuming the field axioms (which are unfortunately named because they are not axioms at all but properties) to derive the definition o the field operators.
Of course, when we say they are field axioms we mean those properties hold true for the elements of the field. If you see those properties, they talk about distributivity over the elements of the field and additive inverses of the elements also belong to the field, so the distributivity automatically applies to additive inverses too.
After that with a little algebra, "product of additive inverses of two elements is equal to the product of the two elements" comes out as a result (not a definition).
Of course, by "product" we mean whatever * represents. It is not necessarily the multiplication operator we see in numbers.
While this product of additive inverses property does follow from the field axioms, there is a short discussion in the blog comments at https://susam.in/blog/product-of-negatives/comments/ which shows that this property holds true for all rings too.
So it is not just numbers for which this property holds true but for all elements of fields and rings too. Quite simply, (-a)(-b) = (a)(b) in all rings where (-a) and (-b) are the additive inverses of a and b respectively.
Like mentioned in another comment on this thread, the assumptions are well known field axioms. They form a good starting point.
And why start with Peano axioms? They seem like a bad starting point because it would take pages upon pages of proof and it won't easily extend to other algebraic structures like rings and fields.
> the assumptions are well known field axioms. They form a good starting point.
I gave Peano as an example. I don't mind the assumptions, as long as they're reasonable and presented before the proof. Another comment pointed me to the fact that they were mentioned in an earlier paragraph, so my issue is resolved.
What your parent comment meant was that it is not possible to learn well from typical blog posts like this that try to condense the subject into a 3000 word article. Of course, if you copy-paste the content of a book into a blog post, then parent comment's point no longer applies.
Off-topic question: I see a lot of new posts coming from .cc websites. What is the appeal that .cc provides? Is it that the content is released under Creative Commons? That's not the case here. What kind of messaging or symbolism or underlying meaning one is aiming for when they are going for a .cc domain?
I doubt it has anything to do with the conceptual meaning of .cc and more with the practical consideration that uxdesign.cc was available and much cheaper than the alternatives.
There are some Creative Commons sites (notable arduino.cc), but mostly it's that .com is basically full, and squatters are sitting on a lot of it. So there's a burst of sites willing to take the hit of being associated with a lower-grade TLD.
Why .cc? I guess because it's familiar-ish. It's easy to hit the "c" twice, and "c" is the first letter of "com".
In this case, uxdesign.com refers to a blog that has lain fallow for a decade. It does no harm sitting there, and it may as well as long as its owner remembers to pay the trivial sum to renew it. But "uxdesign" might as well get new life in some other TLD.
"Spelled" is more common now even in British English. E.g. compare a Google search of 'site:bbc.co.uk "spelt"' with 'site:bbc.co.uk "spelled"'. Most recent usage of "spelt" is referring to the grain, or a direct quote of somebody who prefers the old-fashioned spelling.
Yes, I didn't get the usage of the term "LARP" in this context too. It comes off as rude and condescending to both LARPers and email users. I am not sure the term fits appropriately in this context. All LARPers are aware that they are role playing in a game, not the real world, and they do not pretend otherwise.
To larp as a verb applied to a discussion that is not about actual LARP activities (the primary sense of this word) carries a strong connotation of rudeness.
Etymologically it probably emerged in the infamous *chan-culture, and is in active use there in the sense of ‘pretend play’ or ‘on-line role-playing’. It can be used to signal, in a derogatory way, that you feel that the other is spinning a yarn or indulging in a fantasy, without declaring the fictive nature of their comments, in order to provoke responses and bait others into participating.
I wouldn't use this word in polite conversation, unless you are actually talking about live action role-playing.
I certainly don't intend any offense to LARPers in general, virtually none of whom use encrypted email. Nor, indeed, email users, among whom I count myself! No, I'm pretty specific about where my criticism is aimed.
This is backward reasoning. The chosen definition of multiplication is not to keep things "intuitive". If you start with the field axioms, the chosen definition of multiplication is pretty much dictated by the axioms. If you choose another definition of multiplication, you would end with contradictions like 1 = 0 and such nonsense! And mathematicians abhor contradictions!
"Product of additive inverses of two elements is equal to the product of the two elements" is dictated by the field axioms in all fields.