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Operations On Integers.
The addition of two integers is defined as (a, b) + (c, d) = (a + c, b + d). Hence, for example, (1, 3) + (5, 2) = (6, 5). This will correspond to −2 + 3 = 1.
Actually I think ##n \gt 0## was intended. But this is still not correct because we haven't defined what ## n - 1 ## means. Instead this should read "We define ## 0 = \emptyset ## and we define ## n + 1 = n \cup \{ n \}##", or IMHO preferably* "We define ## 0 = \emptyset ## and we define ## S(n) = n \cup \{ n \}##".jbriggs444 said:I see:
"We define 0 := ∅ and if ##n = 0## then we define n = n−1∪{n−1}."
Surely if ##n \ne 0## was intended.
Can the laws of mathematics be derived from the laws of grammar?symbolipoint said:A very different viewpoint on the titled topic, "What are Numbers" but which misses some of the other specified parts of the title: Numbers are adjectives to help give information about quantity.
I really, really believe no. MAYBE a few people have something different to say.Algr said:Can the laws of mathematics be derived from the laws of grammar?
jbriggs444 said:I see:
"We define 0 := ∅ and if ##n = 0## then we define n = n−1∪{n−1}."
Surely if ##n \ne 0## was intended.
Mark44 said:This is a weird definition, IMO. To compute your example (-2 + 3), that involves adding integers, you have to find pairs of numbers that add up to, respectively, -2 and 3. One possibility would be the sum of the ordered pairs (4, 6) + (8, 5). By your definition the result is (12, 11) = 1.
In short, in order to be able to add -2 and 3, you have to know that (4, 6) represents -2. That is, in order to calculate -2 + 3, you have to be able to calculate 4 - 6.
pbuk said:Actually I think ##n \gt 0## was intended. But this is still not correct because we haven't defined what ## n - 1 ## means. Instead this should read "We define ## 0 = \emptyset ## and we define ## n + 1 = n \cup \{ n \}##", or IMHO preferably* "We define ## 0 = \emptyset ## and we define ## S(n) = n \cup \{ n \}##".
This seems to be quoted from the article https://www.revistaminerva.pt/on-the-nature-of-natural-numbers/ which looks pretty weak to me on a number of points.
* I prefer introducing the notion of the successor because it avoids confusion between ## n + 1 ## as a successor and the addition ## a + b; b = 1 = {0, {0}} ##.
pbuk said:I have some more comments:
symbolipoint said:I really, really believe no. MAYBE a few people have something different to say.
pbuk said:This seems to be quoted from the article https://www.revistaminerva.pt/on-the-nature-of-natural-numbers/ which looks pretty weak to me on a number of points.