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I didn't get very far in this before I encountered this: $27 × 37 = 999$, then the comment

This equality makes sense in the mainstream of mathematics by saying that the two sides denote the same integer and that × is a function in the Cantorian sense of a graph.

Question: What is meant by "...Cantorian sense of a graph"? The article goes on to compare a function with a graph, but does this mean graph in the sense of a Cartesian coordinate system? (I don't even know what tags to give this question.)

Asaf Karagila
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  • Yeah, that is confusion. I’d guess that they are talking about how functions, in set theory, are represented as a graph. In this case $((27,37),999)$ is in a relation, $\times,$ where $\times$ satisfies the conditions for a function. In set theory, we define a function as a certain kind of graph - a set of ordered pairs. But I’m not sure the language here helps. – Thomas Andrews May 17 '22 at 17:28
  • So if $X$ and $Y$ are sets, then a relation from $X$ to $Y$ is a subset of the set $X\times Y.$ A function from $X$ to $Y$ is a relation between $X$ and $Y$ with some additional conditions. We can think of the relation as a “graph” in the same way that we graph a function like $y=x^2.$ – Thomas Andrews May 17 '22 at 17:32
  • @ThomasAndrews What that document is getting at is effectively the difference between the relation definition of a function ("Cantorian," now standard) and the syntactic definition (like $y=x^2$, which iirc is how Bernoulli defined functions way back then). – Kyle Miller May 17 '22 at 17:44

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I took a quick look, and it's the "graph" in the sense of the underlying relation for the mathematical definition of a function, i.e., the complete data table that gives the output for each input. (It's named after graphs, which are pictorial representations of the data. Not every graph can be effectively visualized as a graph, using both senses of the word. Hopefully that confusing sentence clarifies things :-) )

The point being made in context is that the multiplication function has a graph that is infinite (you'd need to memorize the entire multiplication table!), but multiplication can be described by a finite procedure. The first is impractical to store, the second can be encoded in a computer.

This is getting at the idea of what it means for two things to be "the same." In the function-as-a-graph sense, $27\times 37=999$ is true because that's the number in that position of the multiplication table -- you didn't do anything. In the function-as-computation sense, that equality means that if you follow a given multiplication algorithm, that's the answer you'd work out.

These are known as extensional vs intensional points of view, I believe. It has to do whether you ignore the internal (syntactic) representation or not.

Kyle Miller
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