For example I can use multiplication only for calculating areas of rectangles; if so, I would probably hold that “5x3” means “area of a rectangle whose sides measure 5 and 3“, and “there is a real number which multiplied by itself equals two” means “there is a square of area 2”.
Or I can mean “if I add together five groups of three apples each, I would find fifteen objects”.
As a quick aside, I think these two interpretations are actually the same thing in disguise. Areas as measurements have units attached to the numbers. Specifically, the units are squares whose sides measure one “unit length”. So when you’re looking at a rectangle that measures 5x3, you’re noting that there are five groups of three squares (or three groups of five squares, depending on how you want to interpret the roles of the factors). Otherwise it’s hard to see why the area would be a result of multiplying the lengths of the sides.
I think perhaps a better example would be the difference between partitive and quotative division. Partitive (“equal-sharing”) says “I have X things to divide equally between N groups. How many things does each group get?” Quotative (“measurement” or “repeated subtraction”) says “I have X things, and I want to make sure that each group gets N of those things. How many groups will there be?” This is the source of not a small amount of confusion for children who are taught only the partitive interpretation and are given a jumble of partitive and quotative division word problems. It’s not immediately obvious why these two different ideas would result in the same numerical computation; it’s actually a result of the commutativity of multiplication and the fact that division is inverse multiplication. So there’s a deep structure here that’s invisible even to participants that still guides their activities and understanding.
Math is exact in the sense that once the rules of inference are given there is no freedom but to follow them, and unobjectionable in the sense that it is futile to dispute the axioms. Any axiomatic system is like that.
I agree that axiomatic systems are like that, but I don’t think the essence of math is axiomatic. That’s one method by which people explore mathematics. But there are others, and they dominate at least as much as the axiomatic method.
For instance, Walter Rudin’s book Real and Complex Analysis goes through a marvelously clean and well-organized axiomatic-style exposé of measure theory and Lebesgue integration. But I remember struggling with several of my classmates while going through that class trying to make sense of what is “really going on”. If math were just axiomatic, there wouldn’t be anything left to ask once we had recognized that the proofs really do prove the theorems in question. But there’s still a sense of there being something left to understand, and it certainly seems to go beyond matters of classification.
What finally made it all “click” for me was Henri Lebesgue’s own description of his integral. I can’t seem to find the original quote, but in short he provided an analogy of being a shopkeeper counting your revenue at the end of the day. One way, akin to the Riemann integral, is to count the money in the order in which it was received and add it up as you go. The second, akin to Lebesgue integration, is to sort the money by value - $1 bills, $5 bills, etc. - and then count how many are in each pile (i.e. the measure of the piles). This suddenly made everything we were doing make tremendously more sense to me; for instance, I could see how the proofs were conceived, even though my insight didn’t actually change anything about how I perceived the axiomatic logic of the proofs.
The fact that some people saw this without Lebesgue’s analogy is beside the point. The point is that there’s an extra something that seems to need to be added in order to feel like the material is understood.
I’m going to some lengths to point this out because the idea of math as perfect and axiomatic just isn’t the mathematics that humans practice or know. It can look that way, but the truth seems to be more complicated than that.
I think perhaps a better example would be the difference between partitive and quotative division.
Maybe even easier example is the commutativity of multiplication itself. It is not a priori clear that 5 group of 3 objects each are the same as 3 groups of 5 objects each. When I was a child I was feeling confused why addition and multiplication are commutative while exponentiation isn’t.
I’m going to some lengths to point this out because the idea of math as perfect and axiomatic just isn’t the mathematics that humans practice or know. It can look that way, but the truth seems to be more complicated than that.
Yes, we have powerful (sometimes astoundingly powerful, as in case of Ramanujan) intuitions built in our brains that allow us to do high-level operations. Mathematics is practically never done on the lowest level of formal manipulation. There is certainly large difference between mathematics as an axiomatic system and the art of mathematics as a human endeavour—if there weren’t, mathematicians were replaced by machines long ago. But that doesn’t seem much relevant to the question of truth of mathematical theorems. Whatever intuitive thought had lead to its discovery, people will agree that it is valid iff there is a formal proof.
Maybe even easier example is the commutativity of multiplication itself.
That’s a good point! I avoided that example because there’s a pretty easy and convincing “proof” of the commutativity of multiplication, namely that turning a rectangle on its side doesn’t change how many things constitute it So, it doesn’t matter whether you count how many are in each row and then count how many rows there are, or if you do that with columns instead.
I think it’s terribly sad that they don’t encourage children to notice that or something like it. But there are a lot of things about education I find terribly sad and that I’m doing my damnest to fix.
But that doesn’t seem much relevant to the question of truth of mathematical theorems. Whatever intuitive thought had lead to its discovery, people will agree that it is valid iff there is a formal proof.
Agreed, though there’s no objective definition of what constitutes a “formal proof”. Despite what it might seem like from the outside, there’s no one axiomatic system and deductive set of rules to which all subfields of mathematics pay homage.
As a quick aside, I think these two interpretations are actually the same thing in disguise. Areas as measurements have units attached to the numbers. Specifically, the units are squares whose sides measure one “unit length”. So when you’re looking at a rectangle that measures 5x3, you’re noting that there are five groups of three squares (or three groups of five squares, depending on how you want to interpret the roles of the factors). Otherwise it’s hard to see why the area would be a result of multiplying the lengths of the sides.
I think perhaps a better example would be the difference between partitive and quotative division. Partitive (“equal-sharing”) says “I have X things to divide equally between N groups. How many things does each group get?” Quotative (“measurement” or “repeated subtraction”) says “I have X things, and I want to make sure that each group gets N of those things. How many groups will there be?” This is the source of not a small amount of confusion for children who are taught only the partitive interpretation and are given a jumble of partitive and quotative division word problems. It’s not immediately obvious why these two different ideas would result in the same numerical computation; it’s actually a result of the commutativity of multiplication and the fact that division is inverse multiplication. So there’s a deep structure here that’s invisible even to participants that still guides their activities and understanding.
I agree that axiomatic systems are like that, but I don’t think the essence of math is axiomatic. That’s one method by which people explore mathematics. But there are others, and they dominate at least as much as the axiomatic method.
For instance, Walter Rudin’s book Real and Complex Analysis goes through a marvelously clean and well-organized axiomatic-style exposé of measure theory and Lebesgue integration. But I remember struggling with several of my classmates while going through that class trying to make sense of what is “really going on”. If math were just axiomatic, there wouldn’t be anything left to ask once we had recognized that the proofs really do prove the theorems in question. But there’s still a sense of there being something left to understand, and it certainly seems to go beyond matters of classification.
What finally made it all “click” for me was Henri Lebesgue’s own description of his integral. I can’t seem to find the original quote, but in short he provided an analogy of being a shopkeeper counting your revenue at the end of the day. One way, akin to the Riemann integral, is to count the money in the order in which it was received and add it up as you go. The second, akin to Lebesgue integration, is to sort the money by value - $1 bills, $5 bills, etc. - and then count how many are in each pile (i.e. the measure of the piles). This suddenly made everything we were doing make tremendously more sense to me; for instance, I could see how the proofs were conceived, even though my insight didn’t actually change anything about how I perceived the axiomatic logic of the proofs.
The fact that some people saw this without Lebesgue’s analogy is beside the point. The point is that there’s an extra something that seems to need to be added in order to feel like the material is understood.
I’m going to some lengths to point this out because the idea of math as perfect and axiomatic just isn’t the mathematics that humans practice or know. It can look that way, but the truth seems to be more complicated than that.
Maybe even easier example is the commutativity of multiplication itself. It is not a priori clear that 5 group of 3 objects each are the same as 3 groups of 5 objects each. When I was a child I was feeling confused why addition and multiplication are commutative while exponentiation isn’t.
Yes, we have powerful (sometimes astoundingly powerful, as in case of Ramanujan) intuitions built in our brains that allow us to do high-level operations. Mathematics is practically never done on the lowest level of formal manipulation. There is certainly large difference between mathematics as an axiomatic system and the art of mathematics as a human endeavour—if there weren’t, mathematicians were replaced by machines long ago. But that doesn’t seem much relevant to the question of truth of mathematical theorems. Whatever intuitive thought had lead to its discovery, people will agree that it is valid iff there is a formal proof.
That’s a good point! I avoided that example because there’s a pretty easy and convincing “proof” of the commutativity of multiplication, namely that turning a rectangle on its side doesn’t change how many things constitute it So, it doesn’t matter whether you count how many are in each row and then count how many rows there are, or if you do that with columns instead.
I think it’s terribly sad that they don’t encourage children to notice that or something like it. But there are a lot of things about education I find terribly sad and that I’m doing my damnest to fix.
Agreed, though there’s no objective definition of what constitutes a “formal proof”. Despite what it might seem like from the outside, there’s no one axiomatic system and deductive set of rules to which all subfields of mathematics pay homage.