2 apples added to 2 apples yields 4 apples, because that is a physical fact.
2+2=4 because arithmetic was designed to be useful.
If the physical facts regarding apples were to change, 2+2=4 would still be true, but it would not be useful to use the symbols 2 and 4 in the description of quantities of apples.
If the physical facts of apples were to change such that 2 apples added to 2 more apples did not give you 4 apples, then removing the detail that it’s an apple would not yield numbers. In such a case, you would not be able to abstract apples into numbers. They would abstract away into something else.
Likewise, if you changed the mental processes which makes Peano Arithmetic, you would not change numbers; you would merely have changed what Peano Arithmetic can be abstracted into.
The thing to get from my post is that numbers are an abstraction: they are apples, when you forget that it’s apples that you’re talking about. They are Peano Arithmetic, when you forget that it’s mental processes you’re talking about. They are bits, when you forget it’s bits you’re talking about. But this does not make them special, just like the concept “sphere” is not special. We’re just lucky that numbers emerge all over the place in the universe, probably because the laws of physics are the same everywhere so everything is built upon the same base.
You can’t get a number out of a quantity- it’s simply that just about all of the rules that apply to numbers also apply to quantities.
That’s because numbers were created to be useful in manipulating quantities. Mathematics interacts very well with physics from a combination of factors: Physics appears to obey similar underlying laws as mathematics, and mathematics is created by people who want to explain physical phenomena.
When you forget that it’s apples and miles that you are talking about, 2 apples and 2 miles never yields 4 anything, even though 2 apples plus 2 apples is 4 apples, and 2 miles plus 2 miles is 4 miles.
There’s even a term for what you describe in the academic world- Magic Units. It is used to provide partial credit for showing mathematical calculations on numbers rather than showing mathematical calculations on quantities.
Very nice and convincing argument. There were some moments when thinking about it when I though your argument defeated my view. Sadly, we’re not quite there yet.
Trying to add 2 miles to 2 apples does not make sense. There is no physical representation of such an operation. So you can’t try to abstract that into numbers. Here’s an example, to clarify:
Let’s say I’ve got a bag of 2 apples, I add 2, and one falls through the hole in my bag. The number of apples in my bag is 2+2-1=3. The first 2 is an abstraction of the original number of apples in my bag, the second 2 is the number of apples I added, and −1 is the number of apples that fell out through the hole, and 3 is the total. Everything in this equation can be mapped back to reality by adding in details. But in 2 apples + 2 miles, the + cannot be mapped back to reality. And so it is not representative of apples.
I can think of an easy attack on this, by saying that 2 apples + 2 oranges does make sense in our world. But this is a disguised incorrectness. The oranges do not fall into the criterion of “apple”, so it does not actually make sense to add them using the ‘+apples’ operator. Of course, we could generalize the ‘+’ operation to mean ‘+fruit’, but then it becomes 2 fruit + 2 fruit, which does make sense, and is true whether you reject my current views or not.
I end up five miles from where I started, because I dropped units from my quantities and did operations on the numbers.
Or, I could apply a constant force of 5 newtons to an object massing 25 kg for a duration of 8 seconds. I change it’s velocity by 5N8S(1MKG/(NSEC^2))/25KG 1.6 M/S. By conserving units on all quantities, I convert force-time against a mass into acceleration.
Those units can be preserved through all mathematical operations, including exponentiation and definite integration.
I end up five miles from where I started, because I dropped units from my quantities and did operations on the numbers.
Don’t we have the same problem with complex numbers? 2 + 3j = 5
I end up with 5 because I ignored imaginary numbers?
I’m not sure what my point is, this is after all a question. I am wondering, does the fact the same error occurs from dropping physical units as from dropping the very non-physical, seemingly quite mathematical concept of the sqrt(-1)==j?
It’s a different error, because quantities aren’t inherently algebraic, even though they very often behave as though they were.
For example, arranging apples in a grid three wide and two apples deep requires 6 apples, not 6 apples^2, even though the area of a grid two inches wide and two inches deep is 6 in^2.
Or, I could apply a constant force of 5 newtons to an object massing 25 kg for a duration of 8 seconds. I change it’s velocity by 5N8S(1MKG/(NSEC^2))/25KG 1.6 M/S. By conserving units on all quantities, I convert force-time against a mass into acceleration.
Those units can be preserved through all mathematical operations, including exponentiation and definite integration.
Hmm… Another good argument. This one is harder. But that’s just making this more fun, and getting me closer to giving in.
If I abstract a whole bunch of details about apples away, except the number and the fact that it’s an apple, I get math with units. So if I have a bag of 3 apples, I can abstract this to 3apples. I can add 1apple to this, and get 4apples. Why? Some ancient mathematicians have shown that when you retain this extra bit of information, the abstraction “math with units” is formed, which is pretty much exactly like normal math, except that numbers are multiplied by their units. “math with units” and plain old math are very similar because they are both numbers, but “math with units” happens to retain a few extra details about what the numbers are talking about.
1apple 1apple=1apple^2 is true, but it doesn’t make sense. You can’t add in details to turn this into apples because of physical limitations, so 1apple 1apple does not emerge from apples. But it is consistent with the rules of mathematics, which you obtain when you remove the complex details of what an apple is, so this is considered true.
1m * 1m=1m^2, however, does make sense, and we can add in details to transform this into an area. m/s can be transformed into the speed of an object by adding in details. m/s^2 can be transformed into an actual acceleration. kgm/s^2 can be transformed into an actual force by adding in details. I think this is true of all useful units, but if you can think of one for which this is not true, please share, because that will be a big blow to my theory. If this theory survives all the other arguments, I will still need to prove that this is true, and until I prove it, my theory is weak. Thank you for showing me this.
Anyway, a summary of this comment: math with units is an abstraction, one level below plain math, and it mostly follows the rules of math because it’s just math+(some simple other thing). Units which don’t make sense cannot be un-abstracted into things in the world, but because math is consistent, we can still manipulate them like anything else because we have abstracted the details which makes it not make sense away.
So I needed to extend my theory for this, but thankfully not by making it more complex at the base. This is all just emergent stuff in the universe. So if my theory holds up, I expect Occam’s Razor will be in favor of it.
Quantities can be converted to and from numbers: (32 ft lbf / (lbm sec^2))=1 (64 ft lbf / (lbm sec^2))=2
It is true, but not useful, to say that the area of a circle with radius R is equal to piR{frac{64ftlbf}{lbmsec2}}
or equivalently,
Taking the sec^2lbf root does not have an analogue in reality, but the units output from taking the time^2*Force root of a distance raised to the power of scalar*distance*mass is area in this specific case.
Numbers are not relatable to quantities.
2 apples added to 2 apples yields 4 apples, because that is a physical fact.
2+2=4 because arithmetic was designed to be useful.
If the physical facts regarding apples were to change, 2+2=4 would still be true, but it would not be useful to use the symbols 2 and 4 in the description of quantities of apples.
If the physical facts of apples were to change such that 2 apples added to 2 more apples did not give you 4 apples, then removing the detail that it’s an apple would not yield numbers. In such a case, you would not be able to abstract apples into numbers. They would abstract away into something else.
Likewise, if you changed the mental processes which makes Peano Arithmetic, you would not change numbers; you would merely have changed what Peano Arithmetic can be abstracted into.
The thing to get from my post is that numbers are an abstraction: they are apples, when you forget that it’s apples that you’re talking about. They are Peano Arithmetic, when you forget that it’s mental processes you’re talking about. They are bits, when you forget it’s bits you’re talking about. But this does not make them special, just like the concept “sphere” is not special. We’re just lucky that numbers emerge all over the place in the universe, probably because the laws of physics are the same everywhere so everything is built upon the same base.
You can’t get a number out of a quantity- it’s simply that just about all of the rules that apply to numbers also apply to quantities.
That’s because numbers were created to be useful in manipulating quantities. Mathematics interacts very well with physics from a combination of factors: Physics appears to obey similar underlying laws as mathematics, and mathematics is created by people who want to explain physical phenomena.
When you forget that it’s apples and miles that you are talking about, 2 apples and 2 miles never yields 4 anything, even though 2 apples plus 2 apples is 4 apples, and 2 miles plus 2 miles is 4 miles.
There’s even a term for what you describe in the academic world- Magic Units. It is used to provide partial credit for showing mathematical calculations on numbers rather than showing mathematical calculations on quantities.
Very nice and convincing argument. There were some moments when thinking about it when I though your argument defeated my view. Sadly, we’re not quite there yet.
Trying to add 2 miles to 2 apples does not make sense. There is no physical representation of such an operation. So you can’t try to abstract that into numbers. Here’s an example, to clarify:
Let’s say I’ve got a bag of 2 apples, I add 2, and one falls through the hole in my bag. The number of apples in my bag is 2+2-1=3. The first 2 is an abstraction of the original number of apples in my bag, the second 2 is the number of apples I added, and −1 is the number of apples that fell out through the hole, and 3 is the total. Everything in this equation can be mapped back to reality by adding in details. But in 2 apples + 2 miles, the + cannot be mapped back to reality. And so it is not representative of apples.
I can think of an easy attack on this, by saying that 2 apples + 2 oranges does make sense in our world. But this is a disguised incorrectness. The oranges do not fall into the criterion of “apple”, so it does not actually make sense to add them using the ‘+apples’ operator. Of course, we could generalize the ‘+’ operation to mean ‘+fruit’, but then it becomes 2 fruit + 2 fruit, which does make sense, and is true whether you reject my current views or not.
I’m looking forward to your next argument :).
I put two apples in my one bag, then walk two miles and add two more apples to the bag. There is one hole in the bag, and one of the apples falls out.
2 (apples) + 2 (miles) + 1 (bag) − 1 (hole) − 1 (apple) + 2 (apples) = 5
I end up five miles from where I started, because I dropped units from my quantities and did operations on the numbers.
Or, I could apply a constant force of 5 newtons to an object massing 25 kg for a duration of 8 seconds. I change it’s velocity by 5N8S(1MKG/(NSEC^2))/25KG 1.6 M/S. By conserving units on all quantities, I convert force-time against a mass into acceleration.
Those units can be preserved through all mathematical operations, including exponentiation and definite integration.
Don’t we have the same problem with complex numbers?
2 + 3j = 5 I end up with 5 because I ignored imaginary numbers?
I’m not sure what my point is, this is after all a question. I am wondering, does the fact the same error occurs from dropping physical units as from dropping the very non-physical, seemingly quite mathematical concept of the sqrt(-1)==j?
It’s a different error, because quantities aren’t inherently algebraic, even though they very often behave as though they were.
For example, arranging apples in a grid three wide and two apples deep requires 6 apples, not 6 apples^2, even though the area of a grid two inches wide and two inches deep is 6 in^2.
Hmm… Another good argument. This one is harder. But that’s just making this more fun, and getting me closer to giving in.
If I abstract a whole bunch of details about apples away, except the number and the fact that it’s an apple, I get math with units. So if I have a bag of 3 apples, I can abstract this to 3apples. I can add 1apple to this, and get 4apples. Why? Some ancient mathematicians have shown that when you retain this extra bit of information, the abstraction “math with units” is formed, which is pretty much exactly like normal math, except that numbers are multiplied by their units. “math with units” and plain old math are very similar because they are both numbers, but “math with units” happens to retain a few extra details about what the numbers are talking about.
1apple 1apple=1apple^2 is true, but it doesn’t make sense. You can’t add in details to turn this into apples because of physical limitations, so 1apple 1apple does not emerge from apples. But it is consistent with the rules of mathematics, which you obtain when you remove the complex details of what an apple is, so this is considered true.
1m * 1m=1m^2, however, does make sense, and we can add in details to transform this into an area. m/s can be transformed into the speed of an object by adding in details. m/s^2 can be transformed into an actual acceleration. kgm/s^2 can be transformed into an actual force by adding in details. I think this is true of all useful units, but if you can think of one for which this is not true, please share, because that will be a big blow to my theory. If this theory survives all the other arguments, I will still need to prove that this is true, and until I prove it, my theory is weak. Thank you for showing me this.
Anyway, a summary of this comment: math with units is an abstraction, one level below plain math, and it mostly follows the rules of math because it’s just math+(some simple other thing). Units which don’t make sense cannot be un-abstracted into things in the world, but because math is consistent, we can still manipulate them like anything else because we have abstracted the details which makes it not make sense away.
So I needed to extend my theory for this, but thankfully not by making it more complex at the base. This is all just emergent stuff in the universe. So if my theory holds up, I expect Occam’s Razor will be in favor of it.
Quantities can be converted to and from numbers:
(32 ft lbf / (lbm sec^2))=1
(64 ft lbf / (lbm sec^2))=2
It is true, but not useful, to say that the area of a circle with radius R is equal to piR{frac{64ftlbf}{lbmsec2}}
or equivalently,
Taking the sec^2lbf root does not have an analogue in reality, but the units output from taking the time^2*Force root of a distance raised to the power of scalar*distance*mass is area in this specific case.