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The Fourth Operation: What Comes After Exponentiation

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Re: The Fourth Operation: What Comes After Exponentiation

#41

Post by orangecat »

hervature wrote: post #6
paulmooreparks wrote: post #4

There is a bit of a debate about whether or not multiplication should be defined as repeated addition: https://www.maa.org/external_archive/devlin/devlin_06_08.htm... https://www.maa.org/external_archive/devlin/devlin_0708_08.h...

Could not disagree more with this take. Multiplication of fractions is simply the division of two whole number multiplications. Which, if you are teaching fractions, division has already been taught. Seems like a contrarian take for the sake of being contrarian rather than based on pedagogy. Glad he wasn’t my teacher as he would have confused me.

I agree with your disagreement. By this standard we shouldn't teach F=ma in introductory physics, and we should require kindergartners to understand the ZFC axioms before we can tell them what "3" is.

Re: The Fourth Operation: What Comes After Exponentiation

#42

Post by TrainedMonkey »

RHSeeger wrote: post #39
adrian_b wrote: post #34

If it is said that "Those are the first, second, and third operations: addition, multiplication, and exponentiation", then one should not forget that according to this numbering there is a "zeroth" operation: adding 1 to a number (incrementation). Addition is derived from adding 1 to a number in the same way as multiplication is derived from adding a number to zero (if instead of starting with zero the operation is s…

> So the sequence of operations is: adding 1 to a number, addition, multiplication, exponentiation, As I was reading this, I was visualizing it as geometry. - Incrementation as a line, where each "step" moves you along that line - Addition as a 2d graph, where each point along the x axis increments by 1, and each point along the y axis indicates "how many times" - Multiplication as a 3d graph, in the same pattern - E…

I think this is very insightful. Math is designed to tackle problems which are challenging/not possible to visualize. So to get good at symbolic mathematics at some point you need to let go of needing to visualize everything. To get a bit more perspective one could contrast symbolic math with geometry which is built on visualization.

Re: The Fourth Operation: What Comes After Exponentiation

#43

Post by Syzygies »

Several comments independently propose incrementation as coming before addition, multiplication.

One can only make sense of progressions like this by finding relationships between the levels. There are at least two ways to see a connection between addition and multiplication: The logarithm, and polynomials. Polynomials are well-studied expressions combining addition and multiplication.

Algebraic geometry studies sets defined by polynomial equations, as linear algebra studies sets defined by linear equations. Tropical geometry [1] is a version of algebraic geometry that replaces addition by minimization, and multiplication by addition.

Pure mathematicians like to hear music in their choices; they want these choices to arise naturally and support deep theories. Tropical geometry passes this test. One wants an inevitability to one's choices, a belief that alien life would reach the same conclusions.

If there are two answers to "what's zeroth? What's fourth?" that itself is interesting. However, one wants extensive evidence to believe that there's branching. Otherwise, we decide that we've simply stepped onto the wrong trail and need to backtrack. Pure mathematicians live in profound fear of just playing house, making stuff up because it sounds good, even if it looks to everyone else like that's what we do.

[1] https://en.wikipedia.org/wiki/Tropical_geometry

Re: The Fourth Operation: What Comes After Exponentiation

#44

Post by lisper »

From the afterword:

> Multiplication can be considered iterated addition only when one of the numbers – the number measuring the degree of iteration – is pure. Adding five apples to itself three times makes 15 apples. But three apples or three oranges as a count for adding makes no sense.

Hm, I wonder what happens if you consider "times" or "iterations" a unit? I haven't thought this through but it feels like the edge of a deep insight. Playing fast and loose with units and iterations feels like Lisp. Being strict about it feels like Haskell.

Re: The Fourth Operation: What Comes After Exponentiation

#45

Post by hervature »

orangecat wrote: post #41
hervature wrote: post #6

Earlier quoted context omitted.

Could not disagree more with this take. Multiplication of fractions is simply the division of two whole number multiplications. Which, if you are teaching fractions, division has already been taught. Seems like a contrarian take for the sake of being contrarian rather than based on pedagogy. Glad he wasn’t my teacher as he would have confused me.

I agree with your disagreement. By this standard we shouldn't teach F=ma in introductory physics, and we should require kindergartners to understand the ZFC axioms before we can tell them what "3" is.

That's actually the exact opposite of what I'm saying and the exact approach the author is saying. It is unclear what they are proposing, but it smells awfully similar to jumping to modern understanding mathematics in one shot to avoid "repeatedly lying" to students.

As others have pointed out, repeated addition as multiplication readily extends to rational numbers, then to irrational numbers as limits of rational sequences. This is exactly the progression that is taught in Rudin's analysis book and the way to construct the real numbers. At no point in time do you need to backtrack on repeated addition but you need to introduce new concepts division and limits. This is exactly teaching F=ma and then introducing relativity and quantum as the students gain more depth and break past the classical setting.

Re: The Fourth Operation: What Comes After Exponentiation

#46

Post by hervature »

isaac21259 wrote: post #8
hervature wrote: post #6

Earlier quoted context omitted.

Could not disagree more with this take. Multiplication of fractions is simply the division of two whole number multiplications. Which, if you are teaching fractions, division has already been taught. Seems like a contrarian take for the sake of being contrarian rather than based on pedagogy. Glad he wasn’t my teacher as he would have confused me.

What about irrational numbers? There's no neat way to view multiplication of two irrational numbers as repeated addition. And even if there were a way I don't think it's a useful way to think or teach after the first couple years because it makes obvious things like √2×√2 = 2 seem weird and mysterious.

Others have handled the appropriate construction of irrational numbers. The claim here is not that every student needs to know irrational numbers are a limiting sequence but this is exactly how mathematicians think of it. So it is strange to hear another mathematician claim that repeated addition is somehow lying. At some point, people kind of just accept that it is just something you punch into your calculator and don't even think about it anymore.

Your example is quite bad because sqrt(2)*sqrt(2) = sqrt(2*2) = sqrt(4) = 2. So repeated addition works fine. Let's focus instead on pi*pi. The way calculators do this is precisely as some type of limiting sequence depending on how much precision you want. Because, one cannot "calculate" pi*pi exactly because it is irrational. So, you have 3*3, then 3.1*3.1, then 3.14*3.14, etc. which are all repeated additions with some division (e.g. 314*314/(100*100)). In reality, when multiplying two irrational numbers, we just use enough decimal points for floating point precision and then chop off any potentially erroneous digits after the multiplication.*

Re: The Fourth Operation: What Comes After Exponentiation

#47

Post by orangecat »

hervature wrote: post #45
orangecat wrote: post #41

Earlier quoted context omitted.

I agree with your disagreement. By this standard we shouldn't teach F=ma in introductory physics, and we should require kindergartners to understand the ZFC axioms before we can tell them what "3" is.

That's actually the exact opposite of what I'm saying and the exact approach the author is saying. It is unclear what they are proposing, but it smells awfully similar to jumping to modern understanding mathematics in one shot to avoid "repeatedly lying" to students. As others have pointed out, repeated addition as multiplication readily extends to rational numbers, then to irrational numbers as limits of rational se…

Sorry, I was trying to agree with you and apparently phrased it poorly :)

Re: The Fourth Operation: What Comes After Exponentiation

#48

Post by raindear »

adrian_b wrote: post #34

If it is said that "Those are the first, second, and third operations: addition, multiplication, and exponentiation", then one should not forget that according to this numbering there is a "zeroth" operation: adding 1 to a number (incrementation). Addition is derived from adding 1 to a number in the same way as multiplication is derived from adding a number to zero (if instead of starting with zero the operation is s…

The article discusses this. It calls it "succession", and raises arguments against it being the zeroth operation.

Re: The Fourth Operation: What Comes After Exponentiation

#49

Post by SideburnsOfDoom »

RHSeeger wrote: post #39
adrian_b wrote: post #34

If it is said that "Those are the first, second, and third operations: addition, multiplication, and exponentiation", then one should not forget that according to this numbering there is a "zeroth" operation: adding 1 to a number (incrementation). Addition is derived from adding 1 to a number in the same way as multiplication is derived from adding a number to zero (if instead of starting with zero the operation is s…

> So the sequence of operations is: adding 1 to a number, addition, multiplication, exponentiation, As I was reading this, I was visualizing it as geometry. - Incrementation as a line, where each "step" moves you along that line - Addition as a 2d graph, where each point along the x axis increments by 1, and each point along the y axis indicates "how many times" - Multiplication as a 3d graph, in the same pattern - E…

You can visualise incrementation as a single step along a line, addition as a step of any length along the line.

Multiplication: the formula for area of a rectangle is "length times height", so IMHO now this - "a * b" is best visualised as 2d. Or if you like, "m lines of length n", visualised as a stack of lines.

Exponentiation covers "to the power of 2" (area of a 2d object) "to the power of 3" (volume of a 3d cube) and beyond, so now we've run out of graph dimensions.

I think what we can take from this, is that when going from first operation to second, second to third etc, the rate of change is not constant, it is increasing. Since the sequence is "no extra dimensions, one extra dimension, all the extra dimensions"

Re: The Fourth Operation: What Comes After Exponentiation

#50

Post by afiori »

adrian_b wrote: post #34

If it is said that "Those are the first, second, and third operations: addition, multiplication, and exponentiation", then one should not forget that according to this numbering there is a "zeroth" operation: adding 1 to a number (incrementation). Addition is derived from adding 1 to a number in the same way as multiplication is derived from adding a number to zero (if instead of starting with zero the operation is s…

Often in math the property (a + b) * c = ac + bc is more useful than the repeated addition[0]

So if you try to find the zeroth operation in this fashion you get (a ? b) * c = ac ? bc which is satisfied[1] by a ? b = max(a,b) or a ? b = min(a,b).

[0] (a + 1) * c = a*c + c

[1] I do not remember whether these are the only two solutions but with min you get tropical geometry[2]

[2] https://en.wikipedia.org/wiki/Tropical_geometry

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