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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.
The purpose of the table of contents is to provide an organized outline of the content covered in the document "What Are Numbers?". It helps readers navigate through the material and find specific information more easily.
The numbers are categorized based on different topics or concepts related to numbers. Each category is listed with corresponding page numbers to help readers locate specific information within the document.
Yes, the table of contents can be used as a reference guide to quickly find information on specific topics related to numbers. By referring to the table of contents, readers can easily locate the relevant page numbers for further reading.
While the table of contents is not essential for understanding the basic concepts of numbers discussed in the document, it can be helpful for readers who want to explore specific topics in more detail. It serves as a guide for navigating through the content efficiently.
The table of contents benefits readers by providing a roadmap of the document's contents, allowing them to quickly locate information of interest. It saves time and effort by helping readers find specific topics or concepts without having to search through the entire document.