I need mathematical proof that the distance from zero to 1 is the equal to the distance from 1 to 2

Question

I didn't know how to phrase the question properly so I am going to explain how this came about.

I know Math is a very rigorous subject and there are proofs for everything we know and use. In fact, I am sure that if there was anything that we couldn't prove Mathematically, then we wouldn't use it any where in Math or Science.

Anyway, here is the story: Not to long ago I was having a conversation with someone who insisted in telling me that any human knowledge based on Math is a sham, this of course includes Science. He then proceed to tell me that he thought this was the case because according to him there is no proof that zero is a number and second he insisted even if we were to say that zero is a number, that there is no proof that the distance from 0 to 1 is the same as the distance from 1 to 2.

According to this guy, there is no way to measure the distance from something that doesn't exist to something that does. If I understood him correctly he said that is because he just can't see how we can measure from non-existence to something that does exist. In this case, he said that there is now way to know for sure that the length or distance from zero or non-existent number to one would be the same as the distance or length from 1 to 2.

No matter what I said to him, he just disregarded offhandedly. I am so upset that I want to punch him in the face, but I would rather get the proofs and show him that he is completely wrong. To me this would be the equivalent of slapping him on the face with a gauntlet and I would have the satisfaction of knowing that Math and Science are in solid ground and that Mathematicians and/or Scientist haven't pulled a fast one on us.

So please post links to where I can read the Mathematical Proofs for these. Or links that show that to be a fallacy.

Thanks.

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The question was, is there a place where I can get proofs for both of those things. First, for Zero being a number whether an integer, complex or any other kind. And second, for a proof that would show that the distance from 0 to 1 is equal to the distance from 1 to 2.

But I guess what I should really have asked is whether or not these proof exist and where I can find them.

Answer 1

Well, mathematics is based upon axioms--there are some things we can't (and shouldn't) argue about. Take a set, for example... let's just assume we have a naive understanding between us of what a set is: an aggregate/collection of objects. In mathematics, we need axioms from which to develop these "proofs" you're wanting. If we don't have axioms, we have nothing to go off of. Some of these axioms are the Peano Axioms. These are used to construct the natural numbers, sometimes written in a fancy N: $\mathbb{N}$ (0, 1, 2, ...) as far as you can count. Literally, one of the axioms says that if you think you've found the largest element of this set $\mathbb{N}$, add one to it and you've found the next one.

Maybe you're wanting to also know how to define addition? We turn to sets: I have a set of two apples adjoined to a set of five apples is equivalent to a set of seven apples. We just add a "+" to make it easier to write. You can get incredibly complicated with something like:

$f: \mathbb{R}^2\to\mathbb{R}$

$(a,b)\to a+b$

Basically the function takes in two numbers and outputs the sum...

Moving on to the distance thing... you've studied Euclid in your time, so you should remember something about Euclidean Distance. Put simply, the distance between two numbers (in one dimension) is the square root of the sums of squared differences (taken coordinate at a time)... whew. Maybe easier with symbols...

Let's say we have the point 0 and 1 and 1 and 2... their distances are (keep in mind this is 1-D)

$\sqrt{(1-0)^2}$ and $\sqrt{(2-1)^2}$ which are both obviously one. (Notice the square root and the squaring were irrelevant for one dimension.

In our (almost) flat space, we use Euclidean distances. But this distance formula (metric) is just one of the formulas we could use.

Let me know if this helps!

BTW: I'm also interested in what people have to say. My answer can't possibly completely answer this question!

Answer 2

I am so upset that I want to punch him in the face

This would be an understandable (although inadvisable) reaction. Here are a few responses I recommend you give to him:

• "Define what you mean by 'exists'."

• "Define what you mean by '1'."

• "Define what you mean by 'distance'."

• "Prove that 1 exists."

• "Prove that the distance from 1 to 2 is 1."

• "What is the distance from 1 to 1?"

These questions are designed to get him a) actually thinking about what he's saying, and b) quiet.

In response to "Define what you mean by '1'" and "Prove that 1 exists", he will undoubtedly point to "one rock" or "one bird" or "one airplane", etc., and then you can respond, "Ah, but those are all one of something. I am talking about the number 1." Here is where he will be a bit confused. The number 1 is a human abstraction from our experiences with one <object>, but it does not exist as a physical object. Here is my favorite quote regarding this (from Linear Algebra by Fraleigh and Beauregard):

"Numbers exist only in our minds. There is no physical entity that is the number 1. If there were, 1 would be in a place of honor in some great museum of science, and past it would file a steady stream of mathematicians gazing at 1 in wonder and awe."

There are several fundamental issues that your adversary is missing:

Firstly, there is no agreed-upon definition of what it means for something to be a "number". If someone said "I am only going to call even integers "numbers", the odd integers don't count", they are just as correct as someone who calls every mathematical construction humans have ever come up with a "number". Mathematicians have decided on definitions of "integer", "complex number", etc., and the statements that "0 is an integer" or "$\sqrt{2}$ is not an integer" are true because "integer" actually means something. The word "number" is a vague word that has no mathematical content.

Secondly, mathematics is a human construction:

"The mathematician is entirely free, within the limits of his imagination, to construct what worlds he pleases. What he is to imagine is a matter for his own caprice; he is not thereby discovering the fundamental principles of the universe nor becoming acquainted with the ideas of God." - John William Navin Sullivan

"Mathematics is a game played according to certain rules with meaningless marks on paper." - David Hilbert

"[Mathematics] is an independent world/ Created out of pure intelligence." - William Wordsworth

So, what "exists" and what "doesn't exist" depends entirely on what we've set up for ourselves. Mathematicians pick a certain set of axioms, then follow them to logical conclusions. It makes no sense to say, separate from a previous decision about what axioms we're using, that "0 doesn't exist". Only after you've decided what axioms you are using, and what exactly it is that would characterize the object "0", can you decide whether your system includes such an object.

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If you really want to get at him, here is a recommendation: Tell him, "I am going to prove directly to you that 0 exists." For several seconds, pretend to be deep in thought, trying to remember the details of the proof. Surely he will look on with anticipation, eager to attack whatever you are going to present. Then, appear to have recalled the key step, and very slowly and deliberately draw a massive "0" taking up the entire piece of paper. Look extremely pleased with yourself, as if you had made a difficult and convoluted argument.

The fact is, that is exactly what "0", or indeed anything in mathematics, is: a symbol we write down. What "0" means in mathematics can vary widely; perhaps it is the zero element of $\mathbb{Z}$ (the integers), or the function from $\mathbb{R}$ (the real numbers) to $\mathbb{R}$ that is the constant 0 function. Within a given context, e.g. specific axioms and structures, it makes sense to decide what properties an object you want to call the symbol "0" might have (for example, if there is a notion of "addition", you would probably ask for an object "0" to satisfy "$a+0=a$" for all objects $a$), and then, having decided what "0" should refer to if it did exist, actually find out whether it does, within your axioms / structures. But without deciding what "0" means beforehand (and remember, 0 isn't something like "no airplanes", since physical objects are irrelevant), all "0" means is the symbol 0, and it is quite easy to prove that that exists - just draw it!

Of course, this won't be satisfying to your adversary at all, but I guess I'm just trying to think of ways you can get your just revenge by making equally absurd arguments :)

Good luck confronting him, perhaps comment back to let us know how it goes!

Answer 3

I don't think mathematical arguments are going to be of any use here. The guy won't understand them, or he'll deny the foundations they rest on, or he'll raise some other moronic objection just to confuse things. I think the right way to counter someone who says "human knowledge based on Math is a sham" is to ask him whether he uses a computer, a mobile phone, an automobile, an airplane, or any other device more sophisticated than a club, and then point out that those things wouldn't exist if not for Mathematics.

Every time he goes over a bridge or through a tunnel or into an elevator, he is entrusting his life to Mathematics. Every time he takes money from an ATM, he is entrusting his fortune to codes that depend on Mathematics. Jeez, you can't take a step nowadays without depending on some science or technology that wouldn't be there if it weren't for Mathematics. Maybe if you multiply enough such examples, you have some chance of convincing him.

Answer 4

This almost certainly won't convince your friend, but it is one way to think about the rational numbers (and, actually, more than that, but let's stick to that):

Start with a line, in the Euclidean geometric sense (goes on forever in two directions). Pick some point on it and call that point 0; pick another point, not the same point again, and call that point 1. Given those two points, it's possible to find the location of any rational number on the "number line" we've created using geometric constructions—at the simplest level, 2 is defined to be the point on the line that is the same distance and direction from 1 as 1 is from 0.

So, how do we know that the distance between 0 and 1 is the same as the distance between 1 and 2? Because it's part of how we defined 2.

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Back to the number line, continuing to construct natural numbers as 2 was constructed from 0 and 1 is essentially a geometric version of the Peano postulates (mentioned in this answer). Given the natural numbers, the integers can be formed by reflecting the natural numbers across 0 to get negative integers. Addition and subtraction can be thought of as translation, which is geometrically constructible, so we have integers and we can add and subtract them. Dilation by a ratio of two given line segment lengths is geometrically constructible, so we can dilate the point 1 with a dilation centered at 0 and with ratio $\frac{a}{b}$ for any integers $a$ and $b\ne 0$, to get any rational number $\frac{a}{b}$. We can multiply any two rational numbers $x$ and $y$ by dilating one of them with a dilation centered at 0 and with ratio the other one. We can construct reciprocals and define division using them. And so on, building up all the expected mechanics of the rational numbers.

Answer 5

There is quite a deep philosophical point here - well more than one. Mathematics is defined in different ways and the foundational questions are rather subtle. However, we might agree for the purposes of this question that we area dealing with the application of agreed rules of inference to a system of axioms. And our rules should also tell us how we are allowed to make definitions.

Axioms were once conceived of as "self-evident truths", but we now multiply schemes of axioms to provide alternative systems (non-euclidean geometry is just one famous example).

Likewise, rules of inference were once thought of as obvious, but now we have different systems and logics.

One philosophical point is whether reasoning from axioms according to rules of inference increases knowledge - and that depends on what knowledge is and how knowledge is conceived. It increases our understanding of what is true, but is that knowledge, and is it knowledge additional to that conveyed by the axioms and rules of inference?

In order to reason based on the information given, you need to check out what is meant by 0, 1 and 2 - how are they defined, what properties do they have, what rules of inference are allowed? What is meant by distance? These questions are not trivial.

You might also suggest that your scientist takes a ruler and marks two points on it. Then ask him to mark the midpoint between them. Then whether he agrees that the distances between each extreme point and the midpoint are equal, and whether the distance between the extreme points is twice the distance between an extreme point and the midpoint. Then say that you are naming the points 0, 1 and 2 and he has just agreed that they have the properties he is suggesting cannot be satisfied. You have a model which enables you to reason about other points on the same line, extend to a plane etc. and which he is quite free to use in his science.

Answer 6

Look into Abstract algebra, primary definitions of binary operation, neutral elements and groups. Continue with rings and fields. Then go on vector spaces. After some time you should reach metric spaces and norm. Then you will have whole definition of distance, in any dimension.

You can then say to him, in our world with addition, zero is neural element for addition, so it's created by definition. And by definition of metric space (and we live in metric space, where distance is measured by meter, feet, etc) distance between 0 and 1 is same as 1 and 2.

Answer 7

It depends much on what you understand by notation. For example I was thought that $2$ is by definition $1 + 1$ (similary $3 = 1 + 1 + 1$) so in $\mathbb{Z}_2$, $0 = 1 + 1 = 2 + 2$.

As $0$ by similar definition is notation for identity element of $+$ (regardless of what monoid is represented by $+$) and $-$ is reverse operation (therefore forming the group) therefore $2 - 1 = 1 + 1 - 1 = 1 = 1 - 0$.

However it should be pointed that:

• I took one, commonly used, definition of distance. Assuming that by distance you mean metric space there is much more possibilities (for example if you took cities and measure the time it takes to travel between them by car it would form metric space, more or less)
• I took the notation I was thought. It makes number of assumptions - that we take $0$ as identity and $1$ as given (usually either identity of second operation in field denoted usually by $\times$ or generator of one of subgroups). That $2$ means $1 + 1$. That we are using group and measure distance by reverse operation. Depending of what branch you are specialising you may list more and more hidden assumption.

Answer 8

"Against Stupidity the Gods themselves fight in vain." Schiller

With some people, either those who are simply stupid, or those being deliberately obtuse there is simply no way to provide an argument that they will be convinced by. It sounds like in this case your friend takes the view that abstract mathematical objects do not really exist (in whatever ways he defines existence). On that basis there is very little you are going to be able to do to convince him they do. Unless you enjoy this kind of philosophical discussion I'd advise not wasting any more of your time trying to do so.

Answer 9

1. Define the complex number 0 (base 10) : http://us.metamath.org/mpegif/df-0.html

2. Define the complex number 1 (base 10) : http://us.metamath.org/mpegif/df-1.html (Note that the numbers 0 and 1 are constants defined as primitives of the complex number axiom system vs. the definition of the number 2 below)

3. 1 and 0 are distinct : http://us.metamath.org/mpegif/ax1ne0.html Axiom 14 of 25 for real and complex numbers (cf. http://us.metamath.org/mpegif/mmcomplex.html ), derived from ZF set theory (cf. http://us.metamath.org/mpegif/mmset.html#axioms ).

4. Define the unit interval with the Euclidean topology : http://us.metamath.org/mpegif/df-ii.html

5. 0 is an identity element for addition : http://us.metamath.org/mpegif/ax0id.html Axiom 15 of 25 for real and complex numbers (i.e. 0 + 1 = 1)

6. Define the number 2 as 2 = 1 + 1 : http://us.metamath.org/mpegif/df-2.html

QED

Answer 10

As a programmer… one way to explain it could simply be to explain that mathematics is a set of rules defined by humans, and by those rules, you write base-10 numbers a certain way.

Namely how adding one to a base-10 number will increase the last position in the series 0…9. If the last position is 9, it changes to 0 and the second-to-last position is instead increased by one and so on.

So by these clear and simple rules, adding one to 0 gives you 1 and adding one again gives you 2. And subtracting one from 0 would give you -1. The distance is 1 per the rules of this game. As explained eloquently in other answers, you can consider mathematics a human set of rules - perhaps there is no need to go deeper than that.

Answer 11

Balthus: "Zero is a fictional concept!"

Robespierre: "You are right, friend! Mathematicians are fools! By the way, when are you going to pay me the money you owe me?"

Balthus: "I don't owe you any money!"

Robespierre: "Why, certainly; but that's as much as to say, you owe me zero francs; and we just determined that there's no such thing as zero."

Balthus: "Um..."

Robespierre: "Therefore, since you cannot owe me zero francs, you must owe me some other amount. I demand that you pay me!"

In short, rather than arguing from the essential truths (which your friend will ignore anyway), argue from utility. Mathematicians choose axioms based on utility for their current purpose, after all.

More than likely, this person just wanted an all-purpose counterargument for science he doesn't want to believe. If you get rid of that motivation to disbelieve, the person would doubtless abandon their arguments.

Answer 12

Here's a quick thought... The problem is two fold - the proof, and the guy.

Ask the guy how one might define 'stupid'.

"Would someone who didn't know that 1 plus 1 is 2 be stupid?"

Then ask if the two ones in the this sum are the same size. Get him choose which is the one from 0 to 1 and which is the one from 1 to 2...

Actually, if you want a really good proof, from a philospical point of view, try Alfred Whitehead and Bertrand "History of Western Civilization" Russell's "Principia Mathematica" which takes a good 200 odd pages If memory serves to prove that indeed 1+1 probably is 2... (Just don't mention Godel to the guy ;-) )