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This sounds like saying 1+1=2 because somebody decided to make it that way. Whilst in one sense it is true linguistically, in many senses it also isn't true in that the concept itself is real. A multiple of two negatives is a positive is because reality works that way. Whilst someone or a group had to decide it works that way, it was decided it works that way because it indeed can only work that way. It's like the concept of imaginary numbers. (Square root of -1) Whilst the number isn't "real" as in you can count it with your fingers, it is mathematically "real" and exists as a "concept". The concept and its applications are absolutely useful in describing the real world and your computer wouldn't run if people didn't understand imaginary numbers. (Which aren't imaginary at all btw, it's just the name attached and we ran out of words to describe mathematical constructs).
Edit that's a bad example. Take fractions. You cannot count fractions with your fingers. But you can add and subtract them, divide and multiply them and otherwise use them as if they are whole numbers that you can physically count with your fingers, using methods and notations that are fairly easy to do. Now, it is true that fractions exist and everything you do with them are that way because someone or a group of someones or independent, seperated groups decided that fractions work and are described in that way, but ultimately they decided it works that way because that's just how reality operates. You don't cut a cake in half and put it in the fridge and find out that it is now 3/4 of a cake. Unless your uncle ate your cake and brought a new one and ate a quarter of it.
Or take pi. pi was worked out by physically measuring the circumference of a circle and comparing it with its diameter (or radius). Diferent groups in the ancient world all independently "discovered" pi. They all decided what pi was (some rather more accurately than others). Now it is true that pi is what some guy a while ago made it that way, but in reality it can only be that value (or around that value depending on how accurate you want to be) because that is simply how reality is.
Of course you have to question the motive of your uncle. In one sense he is broadening your mind. To question reality. To inquire about the world about you. To introspect on the nature of truth. To set you upon the path of self knowledge that cannot be taken away from you. In another sense he sounds like an idiot. Ask him why 1+1=2 and you will find out which is which.
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Is your unlce Leopold Kronecker?
It is a rather fundamental question wether math is discovered or invented. I do not agree with most of Dangermousecatdogs examples, heck I don't agree with Leopold Kronecker. Math is like the hammer we invented to cut cheese, as in Pi is real, yet in our conventional number system transcendental.
There are different ways to set up math without it contradicting itself. Buzzwords are hyperbolic geometry, rational geometry, Inter-universal Teichmüller theory. I don't know anything about these things, yet in my ignorance I am willing to imagine, that two negatives multiplied does not give you a positive in all shapes math can take.
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Is cheesecake a pie? Can it be a cake? What if I cut the pie in half with a hammer, put it in a fridge and took half away and multiplied it with another negative half pie? Do I recieve a positive quarter pie? I now have 3/4 of a cake?
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On October 05 2017 21:28 Dangermousecatdog wrote:
Is cheesecake a pie? Can it be a cake? What if I cut the pie in half with a hammer, put it in a fridge and took half away and multiplied it with another negative half pie? Do I recieve a positive quarter pie? I now have 3/4 of a cake?
If you multiple two negative half pies, you get a positive quarter square pie.
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On October 05 2017 21:51 Simberto wrote:Show nested quote +On October 05 2017 21:28 Dangermousecatdog wrote:
Is cheesecake a pie? Can it be a cake? What if I cut the pie in half with a hammer, put it in a fridge and took half away and multiplied it with another negative half pie? Do I recieve a positive quarter pie? I now have 3/4 of a cake? If you multiple two negative half pies, you get a positive quarter square pie.
I'm having a little bit of trouble understanding the steps in the example (mainly because a "negative half pie" isn't an actual thing, and I don't understand why we're comparing cakes to pies), so let me try to come up with an example that helps illustrate the "negative * negative = positive" issue. Obviously, it's pretty hard to come up with a concrete example because the act of negating a value (owing, lowering, lessening, reversing, opposing, etc.) is usually a verb rather than a noun. For that reason, it might not make sense to assume that we can take a negative tangible object and multiply it by another negative tangible object, but we can still describe a process where those negative labels might make sense in context.
- The act of removing*** an object, for example, can be represented as negating that object. If we take the reverse*** of that decision, we're adding an object. Therefore, we have a double negative which serves the same purpose as a positive.
- The opposite*** of a temperature decreasing*** means that a temperature is increasing.
- If I owe*** you money and then the debt becomes reversed***, then I'll be receiving money from you.
***All of these terms can be represented as a negative sign in some context, computationally or conceptually.
Therefore, we may choose to represent these situations as a double-negative. While they may be "simplified" into a positive overall, retaining the double negative may actually help describe the two-part negation context that would otherwise be lost by merely seeing a positive.
The same line of reasoning works for whether it's establishing that x - - y = x + y or that -x * -y = xy, because once we understand that a double-negative nets you a positive, the latter can be rewritten as -1*x*-1*y = (-1)(-1)(xy) = (1)(xy), unless someone has a further issue with multiplication or commutativity.
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A geometric argument:
Multiplying by a positive number represents stretching or shrinking.
Multiplying by a negative number represents stretching/shrinking together with a 180* rotation.
So if you multiply by two negative numbers, the two 180* rotations make a 360* rotation = no rotation. You are just left with the stretching/shrinking, which is a positive number.
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how about this
we know that
-1+1 = 0
and that
-1*0 = 0
through substitution we have
-1*(-1+1) = 0
now distribute
(-1*-1) + (-1*1) = 0 -----> -1*-1 + (-1) = 0 ----> -1*-1 = 1
for purposes of a proof I suppose that you could take any 2 negative numbers that are to be multiplied
say -a*-b = x
and change it to the form
(-1)*(-1)*a*b = x
and through the above we have already shown (-1)*(-1) = 1
and so we have 1*a*b = x
would this suffice as a proof?
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I don't know if that would be a formal proof per se, but that's what I wrote above (the factoring out of the two -1's) so I definitely think it's a pretty decent justification.
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On October 06 2017 00:29 Day_Walker wrote:
A geometric argument:
Multiplying by a positive number represents stretching or shrinking.
Multiplying by a negative number represents stretching/shrinking together with a 180* rotation.
So if you multiply by two negative numbers, the two 180* rotations make a 360* rotation = no rotation. You are just left with the stretching/shrinking, which is a positive number.
No, that's silly, that's like saying the reason the earth moves the way it does is because shadows on earth have to look the way they do - you have it the other way and it's a consequence of more fundamental things.
You're better off thinking about multiplying by a constant moving along a line in a x-y coordinate system, and you can stretch at any point on this line (going to infinity on both sides)
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minus 1 squared is +1.
(-1*-1) + (-1*1) = 0 -----> -1*-1 + (-1) = 0 ----> -1*-1 = 1
so you have
1-1=0 and 0=0
On October 05 2017 20:15 Dangermousecatdog wrote:
This sounds like saying 1+1=2 because somebody decided to make it that way. Whilst in one sense it is true linguistically, in many senses it also isn't true in that the concept itself is real.
1+1=2 is merely an abstraction with no meaning in the real world.
however,
if i have 1 apple and then i acquire a 2nd apple.. i now have 2 apples.
there is the real world application of 1+1=2.
also,
"pi" isn't really "math". "pi" is a constant.
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On October 06 2017 09:57 FiWiFaKi wrote:Show nested quote +On October 06 2017 00:29 Day_Walker wrote:
A geometric argument:
Multiplying by a positive number represents stretching or shrinking.
Multiplying by a negative number represents stretching/shrinking together with a 180* rotation.
So if you multiply by two negative numbers, the two 180* rotations make a 360* rotation = no rotation. You are just left with the stretching/shrinking, which is a positive number.
No, that's silly, that's like saying the reason the earth moves the way it does is because shadows on earth have to look the way they do - you have it the other way and it's a consequence of more fundamental things.
Right, in this case we get to choose how the earth moves, and we can make our choice based on the shadows we want to see. The same thing is actually going on in the algebraic proof that travis gave: we want certain properties (shadows) like distributivity, and for those properties to hold we see that -1 * -1 must be 1 (the earth must move a certain way).
On October 06 2017 09:57 FiWiFaKi wrote:
You're better off thinking about multiplying by a constant moving along a line in a x-y coordinate system, and you can stretch at any point on this line (going to infinity on both sides)
I'm not sure I understand you here. Are you looking at the graph y = kx? What do you mean by stretching at a point?
The reason I like thinking about rotations is that it fits with arithmetic in C. For example if we view -1 as a 180* rotation, it makes sense that -1 should have two square roots: 90* rotation clockwise, and 90* rotation counterclockwise.
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On October 06 2017 10:07 JimmyJRaynor wrote:
"pi" isn't really "math". "pi" is a constant. Sure, and the entire branch of geometry isn't maths either.
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I think his argument is along the lines that pi is only a number. Saying a number is maths is slightly weird, but not entirely. I would say that "3 is maths" doesn't sound quite right. Obviously numbers are a part of maths, and if you get into the definition of natural numbers via peano axioms, it gets quite mathy. But i still don't think "3 is maths" is a very sound statement.
Basically the same goes for pi. Pi is something you use in geometry, and geometry is maths. But pi itself is just a number.
Granted, the whole rant isn't very exact and more feelings-based, but i can understand why one would say that pi isn't maths.
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I interpreted it the same way, and I agree too.
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Pi itself is not just a number. All constants are mathematically interesting. All constants are "maths". It's like saying e is just a number. It's a number with meaning and is one of the foundations of trig and geometry. It's a number that has to be calculated using algorithmic calculations. It is also a good example or something that at face value is because some guy made it that way, but if you look into it deeper it is both true and not true at the same time, just like multiplying two negatives make a positive.
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On October 05 2017 12:04 bo1b wrote:
This is probably below everyone's pay grade, but is there a proof that two negatives multiplying make a positive? It seems obvious to me that the opposite of an opposite is the original, yet my uncle insists that the entire world operates on negatives multiplying out to positives because some guy a while ago made it that way. Any help in convincing a crack pot would be appreciated.
A mathematical proof only works if you have defined your axioms. As was mentioned before a completely rigorous proof would involve the construction of integers and the respective definition of the order relation (starting from the axioms of some set theory) which would be overkill right now.
Instead we take the following rules (and the typical rules of calculating) as axioms themselves, which I doubt your uncle would take issue with:
For all integers (or rational numbers or real numbers) a,b,c it holds
A0 either a<b, a=b or b<a
A1 (a<b and b<c) implies a<c
A2 a<b implies a+c<b+c
A3 (a<b and 0<c) implies ac<bc
Statement 1. c<0 implies 0<-c
Proof: c<0 implies by A2 0=c-c<0-c=-c
Statement 2. (c<0 and a<b) implies bc<ac
Proof: c<0 implies 0<-c by Statement 1. It follows from A3 -ac<-bc. Now add ac on both sides (via A2) to arrive at 0<ac-bc. Finally add bc on both sides (again via A2) and arrive at bc<ac.
Corollary (a<0 and c<0) implies 0=0*c<ac.
Another nice corollary is 0<1:
Assume this is wrong, i.e. 1<0 (because 1=0 is excluded). Then a<b implies 1*b<1*a by Statement 2 which is a contradiction of A0.
[EDIT]: I just realized that it's nicer for your purpose if you substitute A3 with
A3' (0<a and 0<c) implies 0<ac
It shouldn't be hard to convince anyone that the product of two positives remains positive. From A3' you get A3 in the following way:
a<b implies (by A2) 0<b-a which implies (by A3') 0<bc-ac for 0<c from which it follows ac<bc (again by A2).
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Pi and its relation to a perfect circle illustrates neatly why it's only tangentially related (yet still very useful) for our understanding and applications of nature.
You can't create a perfect circle in reality, you'd need infinite precision for that (infinitesimal small sizes), which reflects neatly in the constant.
I guess a similar story for "e" can be used too, but I haven't thought about it nearly enough and its uses are much more esoteric to me so no claim about that one!
I'm not trying to diminish the beauty of math, by the way, just trying to argue how man made (or how conceptual) maths actually is. It's more of an abstract thing than a real thing imo; and that in itself is awesome if you think about it.
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Random linguistic question, after reading everyone's preferences:
In the United States, we typically abbreviate "mathematics" as "math", but I've noticed that people from many other countries prefer "maths", often times as a singular noun. I can understand if someone says something like "The different maths you might explore in high school are algebra, geometry, trigonometry, and calculus" - implying that maths is plural and responds to multiple branches of mathematics - but the idea that "pi is maths" or "algebra is maths" or "this is an example of maths" as a singular noun is foreign to me. I was just wondering if there was any additional nuance as to why some people prefer "maths" over "math" when referring to a singular entity. Thanks 
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Northern Ireland22208 Posts
On October 06 2017 21:43 DarkPlasmaBall wrote:Random linguistic question, after reading everyone's preferences: In the United States, we typically abbreviate "mathematics" as "math", but I've noticed that people from many other countries prefer "maths", often times as a singular noun. I can understand if someone says something like "The different maths you might explore in high school are algebra, geometry, trigonometry, and calculus" - implying that maths is plural and responds to multiple branches of mathematics - but the idea that "pi is maths" or "algebra is maths" or "this is an example of maths" as a singular noun is foreign to me. I was just wondering if there was any additional nuance as to why some people prefer "maths" over "math" when referring to a singular entity. Thanks
i think it's just because the full word is 'mathematics' which sounds plural, and so the abbreviation 'math' should naturally also be plural
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On October 06 2017 22:08 ahswtini wrote:Show nested quote +On October 06 2017 21:43 DarkPlasmaBall wrote:Random linguistic question, after reading everyone's preferences: In the United States, we typically abbreviate "mathematics" as "math", but I've noticed that people from many other countries prefer "maths", often times as a singular noun. I can understand if someone says something like "The different maths you might explore in high school are algebra, geometry, trigonometry, and calculus" - implying that maths is plural and responds to multiple branches of mathematics - but the idea that "pi is maths" or "algebra is maths" or "this is an example of maths" as a singular noun is foreign to me. I was just wondering if there was any additional nuance as to why some people prefer "maths" over "math" when referring to a singular entity. Thanks i think it's just because the full word is 'mathematics' which sounds plural, and so the abbreviation 'math' should naturally also be plural
Oh... but just because a word ends in "s" doesn't mean it's plural lol. Mathematics is almost always a singular noun, not plural... mathematics is the study of many topics, but there's no such word as "mathematic". My favorite subject is mathematics, not my favorite subjects are mathematics. Oh well.
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EPT
