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I just came across this statement when I was lecturing a student on math and strictly speaking I used:

Assuming that the value of $x$ equals <something>, ...

One of my students just rose and asked me:

Why do we assume so much in math? Is math really built on assumptions?

I couldn't answer him, for, as far as I know, a lot of things are just assumptions. For example:

  • $1/\infty$ equals zero,
  • $\sqrt{-1}$ is $i$, etc.

So would you guys mind telling whether math is built on assumptions?

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Like everything else. –  ABC Nov 5 '13 at 16:43
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Well, yes...as anything else is, right @Anz Joy? I mean, we build houses on the assumption that our instruments are accurate enough and cement, bricks, etc. will remain where they are according to our calculations, and we know the sun won't explode in the next few billion years assuming our calculations and measurements are accurate enough, and we walk on our city's streets under the assumption that the probability that a huge hole will open in front of us and swallow us is as high as a standard politician's IQ, etc. –  DonAntonio Nov 5 '13 at 16:43
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I think the answer is: It is complicated. At the turn of the 20th Century people were building systems of axioms (that is, assumptions) to try and formalise what they saw in every day life. The Peano axioms formalise the natural numbers, and so on. However, Kurt Goedel proved that this approach wasn't as solid a foundation as people had hoped, so I understand that people who work in the foundations of maths have been looking elsewhere. But I don't know. I am a group theorist. Four axioms rule my life. –  user1729 Nov 5 '13 at 17:03
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$i = \sqrt{-1}$ is a definition, not an assumption. –  Soham Chowdhury Nov 5 '13 at 18:43
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I really don't get this student's complaint, or your confusion with it. Why does "assume $x=5$" count as something more important than assumptions in any other hypothetical story? "assume we have a child in a red cape walking to grandma's house" is doing the exact same thing. It's not an assumption about one reality, but one needed to set up the story we want to tell. –  Robert Mastragostino Nov 5 '13 at 20:25

22 Answers 22

To respond to the charge that "we assume so much in math": math involves the analysis of various hypotheticals. When I say "if X then Y," I might need to assume X hypothetically while in the process of proving the implication, but this does not require presumptuousness in any way. One may easily accept "if X then Y" and its proof without actually accepting X at all!

On the other hand, there are various configurations of axioms that can be chosen as the logical foundation of mathematics, with ZFC as the tacit standard. Outside of foundations and set theory and model theory and logic and so on, the axiom list is relatively small (compared to the whole of the theory of any mathematical context with all of the various theorems and formula) and more importantly unchanging, so this is not relevant to the charge "assuming so much in math."

And further away from presumptuousness, many things in math (as anywhere else) are not actually "assumptions" but are conventions and definitions. I don't "assume" an apple is a red or green fruit that grows on trees, for example, that's just how it's defined.

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I like this answer because it deals both with the fact that things are built on a framework of axiomatic assumptions, and with the fact that the word "assume" begins so many statements. The former might be alarming until you get used to it, but the second isn't alarming at all as long as you believe that whatever you are assuming will often be the case! –  Matt Pressland Nov 5 '13 at 18:00
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Nice answer. There is a stray "ness" (or "being") in the last paragraph though. –  Trevor Wilson Nov 5 '13 at 22:01
    
and, given a few assumptions, many, many, many, many things are after that constructed/proved. Assumptions are just a tiny part of maths. –  Olivier Dulac Nov 6 '13 at 17:10

The basic assumption of the "working mathematician" is the following: The logical and set-theoretical environment of his deliberations is not contradictory.

When we say: "Assume the triangle $ABC$ has a right angle at $C$" then this is not an assumption about the "universe of truth", but the announcement that we are talking about right-angled triangles in the sequel.

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But even then there are dialethists . . . –  Shaun Nov 5 '13 at 17:08
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Actually, you don't even need to make that assumption in almost any proofs, except maybe relative-consistency proofs, since if your logical environment is inconsistent, then the thing you are trying to prove is true. Thus the possibility that your foundations are inconsistent does not hinder your proof, and you don't need to assume it in any standard proof. –  Mario Carneiro Nov 13 '13 at 14:09
    
@MarioCarneiro Elegant! –  Nikhil Panikkar May 16 at 5:44

In my very first class in the university, Linear Algebra I, the professor began by telling us the following thing which I have engraved deep into my memory:

Mathematics is a science of deductions. We assume certain things, and we infer conclusions from them.

Later I had learned that we also assume that our inference rules are sound, and that the foundations are solid. Of course we can't really prove these things in full, as that would amount to verifying infinitely many things, which we can't do. We always have to assume some consistency of the system we work with. Of course, we don't just randomly assemble rules and assume that they work, we have more than a few centuries where mathematics was based on "natural deductions", so to speak, that allows us to try and distil the assumptions we want to work with.

This is one of the arguments against mathematicians that formalists, or people who see mathematics as a formalism, take: it's just pushing symbols on a piece of papers, and it's all just vacuous because it's a lot of assumptions which rarely have something to do with reality.

But then again, can we really prove that yesterday happened? that we are alive? that we exist? Not in the full, mathematically rigorous, and fully convincing sense of the word. But we have some coherence, some consistency, and we are easily convinced of our own existence, and that it did not began yesterday. Even if it's all lies, and you're all in my head -- you wouldn't believe me if I told you that.

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I think a better way to say this is that nothing is assumed in math. Nearly every mathematical statement is really saying: "if this is true, then this is also true".

Certain assumptions are so common that they are usually left off (suppose that $\mathbb{R}$ is a set of elements which has the following properties... ), but you should never make the mistake that anything is assumed by definition.

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Why "nearly" every statement? In the end, we just say what we can infer from the axioms. –  Christoph Nov 5 '13 at 16:52
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I think you're right, I had left the 'nearly' off originally, but felt like I was going to get yelled at by a logician. –  Cheyne H Nov 5 '13 at 16:53
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This is, of course not true. You assume that you can make inferences. Do not mistake that the axioms, or hypotheses are the only assumptions in mathematics. The very way you play with them is also assumed. –  ABC Nov 5 '13 at 17:36
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ABC: That doesnt invalidate his argument, its still of the form "If (this and these inference rules) then that." –  Abdulh Khazzak Gustav ElFakiri Nov 5 '13 at 18:22
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Inference rules are just axioms. You dont have to believe in them any more then the other axioms to perform math. –  Abdulh Khazzak Gustav ElFakiri Nov 5 '13 at 21:44

$\frac{1}{\infty} = 0$ is not an assumption, it is a convention. $\sqrt{-1} = i$ is just the symbolic representation. 'i' just represents $\sqrt{-1}$. When we start proving a theorem, we assume required hypothesis and try to get some new results by logical steps, so we assume something as a hypothesis in every theorem.

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I would go further; $\frac{1}{\infty}=0$ is not an assumption, its a falsehood... ;) (Or, more accurately, it's not really a well-defined statement). –  Matt Pressland Nov 5 '13 at 17:58
    
@MattPressland: It’s an assumption in some systems, a falsehood in others; the point being that we rarely make absolutive statements (x is true) and more often implicative statements (y is true iff x is true). –  Jon Purdy Nov 6 '13 at 8:34
    
@JonPurdy I can believe there is some system in which this makes sense and is true; fair enough. I guess I had some hidden assumptions of my own that we were not working in such a system. –  Matt Pressland Nov 6 '13 at 10:11
    
I agree with Matt; $1/\infty$ equals zero is incorrect. $1/\infty$ strives to zero is correct (something like $1/\infty \approx 0$). edit The way @Thomas wrote it is the preferred way. –  Heliac Nov 6 '13 at 11:13

I think there are a few interpretation of the question "why we assume so much in math". I would answer some of them follows:

  • Why mathematical statements assume so much?
    • They don't – the premises are implicitly a part of the statements, so the statements themselves don't assume anything.
  • Why the premises of mathematical theorems are so huge?
    • Because the proof wouldn't work otherwise. (Or "because the proof of a more general theorem is too complex to be covered within a single class" or yet something else appropriate for the context.)
  • Why we assume the basic logic (here: sentences like "if $X$ then $Y$" and other similar) to work like we expect it to?
    • It isn't only logic and you do it almost all the time, e.g. when thinking or communicating with others. We assume certain properties of our (i.e. not necessarily "real") world, properties of what we are using to transfer knowledge (e.g. natural language) and properties of agents involved in that process (i.e. ourselves and other people). We usually expect the transfer to reach the target (e.g. acoustic waves to reach the ear), the other person to hear what we would hear (e.g. the sounds, words, etc.) and to interpret the message closely to how we interpret it. Sometimes this fails and we aren't being understood. Without those assumptions we couldn't function like we do (e.g. we would lose most of our evolutionary advantage of having big brains).
  • Why we assume so much about our world?
    • Because without it we wouldn't be able to form any useful predictions. For example we know that what Newton thought about gravity isn't true, but it is still a useful model even today.

I hope this helps $\ddot\smile$

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@ABC Your edit was not aligned with what I had in mind writing the post, it changed the meaning of previous bullets and I find such edits kind of rude (that wasn't a small fix like a typo and you didn't even left a comment asking if this is ok). –  dtldarek Nov 5 '13 at 18:04
    
I don't think we assume basic logic in math in the same sense as the other things you mentioned. Logic aren't facts that's simply "out there for us to discover" (in fact there are many types of logic). They are also part of the axioms that are used in math. So I don't think you should compare it to us assuming that the other person will hear what we hear etc. –  Pratyush Sarkar Nov 6 '13 at 2:16
    
@PratyushSarkar Even if you form the logic as axioms, you need a language in which you will do it (meta-language), because there is nothing yet for you to use. Every time you define something from scratch, you need to assume something about the method you are using to describe your model/theory/axioms/whatever. And this is usually the place where the assumption I was talking about in that bullet point go. It's like talking with aliens – you need some common frame of reference – and with that frame come assumptions. –  dtldarek Nov 6 '13 at 7:58
    
maybe add that in proofs, we assume the premises to be true just to make our life easier - otherwise the whole premises need to be carried around the whole time. Case distinction is similar, as well. (Or making an assumption is a form of case distinction - if the premises are false, then the whole statement is true, at least according to the logics i know of) –  kutschkem Nov 6 '13 at 10:15
    
@kutschkem You are right, but there are many subcases of this (like the "case distinction" you have mentioned), and I've put that all under the hood of "the premises are implicitly a part of the statements" to keep it simple. Thanks for suggestions, though! –  dtldarek Nov 6 '13 at 10:41

Short answer is: No, mathematics is not built on assumptions.

First: A lot of times we will write things like "assume that $x$ equals ". And, a lot of times, we simply just mean that if $x$ equals something, then .... That is, we are saying that what every follows is valid when $x$ is . So if you ever find yourself in a situation where you actually have $x$ equal , then you can apply the conclusions.

Second: Now that $\frac{1}{\infty} = 0$ is not an assumption. It is just wrong. For this to even make sense, you would have to allow dividing with infinity and we usually don't do that (especially in an undergraduate calculus class). Infinity if not a number and we can't divide by it.

What we do have is that $$ \lim_{x\to \infty} \frac{1}{x} = 0. $$ An, unfortunately, many teachers will write that this limit is actually equal to $\frac{1}{\infty}$.

Third: About $\sqrt{-1} = i$. This is not an assumption, but a definition. Having the real numbers, we construct the complex numbers. Formally you can define the complex numbers as the set $$\mathbb{C} = \{a + bi \mid a,b\in \mathbb{R}\}. $$ Here we can take $i$ as a symbol and we define the usual addition/subtraction/multiplication/division with the convention that $i^2 = -1$. This makes $\mathbb{C}$ into a field. We could also express that $i^2 = -1$ by writing $\sqrt{-1} = i$. So, it is all about definitions, not assumptions.

Forth: If we are just making assumption in mathematics, my questions would be: what are we assuming? Are we assuming something about reality? Some will point out that "mathematics starts with axioms, so we are starting with something that we don't prove, therefore we must be making assumptions". I would say that this is an unfortunate use of the word assumption. By saying that axioms are assumptions, you are assigning a meaning/value to the terms used in the axioms. That is, you are saying that the terms here are already well-defined and that we are making well-defined statements about them. The problem, if this was the case, is that those statements could be false or true. And axioms are taken as true. They are more like definitions then assumptions. Note also that we don't just have axioms, but we also have defined the rules we use to deduce statements from the axioms.

Fifth: So how would I answer your student? First I would point out the above, and then I would tell the student to think about mathematics as a game. When you play a game you first establish the rules. Then you play the game. It doesn't make sense to question the rules. The game is defined as the rules. You can ask whether or not the rules are consistent, but you can't really ask if by having the rules we are making assumptions about the world or reality. This approach of course cuts some of the connection between reality and mathematics. There are, of course reasons for the choice of axioms. And so maybe the good questions is: Why did we take those axioms and not other ones? Some (see comment below) will point out that the choice of axioms is like picking up the right tool needed. But if you ask that question some mathematicians will claim that you have moved outside the area of mathematics and into philosophy.

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Why the downvote? –  Thomas Nov 5 '13 at 22:26
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I like your answer, but perhaps a more satisfactory response to "Why these axioms and not others?" is because of their usefulness? It's like picking up the right tools to work with--if your goal is to cut a steak (solve a particular kind of problem), you'd better use a knife instead of a spoon (a particular set of axioms instead of another.) –  Heisenberg Nov 6 '13 at 0:52
    
@Anh: Thanks for your comment. I agree with your point. There is, of course, still the next question then of how we know what tools are useful. I would probably say that this ends up being more of a philosophical/historical discussion. –  Thomas Nov 6 '13 at 1:43

Mathematics is founded on axioms; in a sense they are assumptions, but in a sense they are more than that, because they are explicit.

But when your student raised his hand, you were using "assume" in a different sense: You were making a conditional statement: "Assuming that the value of $x$ equals ..." means that what you are about to say only follows if $x$ has that value; it does not mean that $x$ has the value in question, or that you believe it to. So, it is very different from letting $\sqrt{-1} = i$, which is a definition, or from a convention like not calling $1$ a prime.

So the main reason your students hear the word "assume" so much is that it has a lot of different uses.

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Good point about the various senses of "assume." –  LarsH Nov 5 '13 at 20:17

Yes, very much so.

Usually when we do mathematics at a basic level we do not state the assumptions explicitly because objects like the natural numbers, the reals, logical reasoning etc. are very familiar to us, but to do mathematics formally we must have some basic assumptions behind it all - these are our axioms.

The natural numbers are axiomatised by the Peano axioms. If we didn't have this list of assumptions, how would we ever reason with these objects? Any kind of reasoning has to have a starting point from which other statements are deduced. I think this is the ultimate answer to your question; mathematics is about logical deduction and without some assumptions, where would are deduction begin?

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I think it is extremely important to be aware that the charge "we assume so much in maths" is probably not directed at or in response to foundations or axioms, but rather about the observation that often mathematical discussions begin with words like "assume X,Y,Z." –  anon Nov 5 '13 at 17:03
    
Is the logical deduction we adopt absolutely applicable to other possible creatures in the universe or do they have different ones that might be illogical for us? –  cyanide-based food Nov 5 '13 at 17:10
    
@Marienplatz If you accept the notion that pure logic doesn't work in other physical locations as a real possibility, then you effectively don't believe logic holds at all, and you might as well believe everything and nothing and find all intellectual exercise whatsoever pointless. –  anon Nov 5 '13 at 17:16
    
@anon That is why I added the final sentence; assumptions are the starting point for reasoning. –  Harry Wilson Nov 5 '13 at 17:18

Any mathematical objects that may seem unconnected to reality, weren't being dreamed of a hundred thousand years ago. Modern mathematics is one of the pinnacles of social evolution and it can all ultimately be traced back to the jungle. If mathematics makes any sense at all it is somehow, however unclear at the moment, if it was built by following one logical step to the next, connected to the real world.

The discovery and early work with matrices is a good example. Very abstract initially but led to many important applications approximately a hundred years later. Many of the abstract paths we have gone down began with a desire to find better ways of proving results and solving certain problems that were themselves initially real world problems and have often succeeded in just that. Sometimes there's an element of losing ourselves in abstractions only to find ourselves with a better understanding of reality at a later stage. And Einstein, using some of Minkowski's abstract mathematics, solved a hugely significant real world problem for which previously used mathematics models were inadequate. Did Minkowski expect that when he developed that mathematics?

Historically, there was a reality before any assumptions about it existed. Simply put, humans proceed in the world by making, and testing assumptions. The progress they make is a reflection of the voracity of their understanding. But, that doesn't make reality work. Reality works, however reality does, we just discover how it works.

This is not to say that our actions don't change what exists in reality, but we don't change the how, by which I mean, for example, we can't change the laws of physics.

The original statement is about maths being built on assumptions. Well it's not just assumptions, but assumptions that turned out to be correct. Once they're demonstrated to be correct, they're not just assumptions any more. The Sydney harbour bridge is not built out of assumptions, it's built out of the real stuff that we assumed correctly about.

Even the purest of maths, no matter how seemingly removed from reality is not built on arbitrary assumptions, it still "works" somehow, we don't just make it up as we go along. However complex a structure maths has become, it is connected to the way we think, it is in fact in large part a model of how we think and, as history has progressed, its increasingly complex twists and turns reflect how our thinking has evolved. Underneath all that we strive for it to make sense, even if it may not have an obvious real world application at the time.

Thinking is real. It takes place inside real existing minds. So, my answer is a resounding no, it's not all based on assumptions it's just the pinnacle of how we have come to think.

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Maths is built on assumptions in that the data and logic function correctly.

The utility of mathematics is that it does not reflect some truth of nature as such, but that if certain conditions apply, then these outcomes will happen too.

For example, while we think of Euclidean geometry as applying to straight lines on a plane, it holds also for circles passing through a point. That is, if we call a circle through a point a 'straight line', and parallel straight lines become circles that are cotangent at that point, then we need only demonstrate the postulates to demonstrate the geometry holds.

Another utility comes that algebra can be extended to non-numerics. That is, we can let variables be whole equations or matrices, and still do the algebra.

Once one understands a particular case, one can liken it to a known mathematical problem, and convert its solution into something that applies for this case. I used the various edges of the fano triangle to create a 'perfect game' for eight golfers, such that they play with each member once and against the same member twice (in two pairs).

The general utility is that if one correctly guesses the principles of something as being a known mathematical structure, it is easy to pull answers relative to that something.

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While there is already a plethora of answers to this question, I want to point to something:

In philosophy, we either make our assumptions clear, or make arguments that are too ambiguous to be meaningful

When one considers anything in math, one is considering the subject, nature, and implications of a mathematical statement or mathematical object. However, if we do not specify any assumptions then all of math can be invalidated by a simple argument: "You can not absolutely, ontologically, prove that anything is real, therefore you can not prove mathematics is real." If we make the implicit assumption that there is reality, then we can further develop ideas within reality.

This sort of problem lies at the center of all ontology. We can not effectively describe, discuss, and theorize about objects without discussing and agreeing on what objects are. Because in mathematics we want our logic to be strict, and because we want a lot of generality and abstraction, we assume deductive logic works because it does not lead to contradictions (how true this logic is, is a matter of assumption as well). Furthermore, we say that we can define objects, we say that these definitions are valid for a particular situation if they do not lead to a contradiction within our logic that we assumed worked.

However, the mathematics itself can follow these assumptions but be meaningless. This is why we have certain axioms, which are a combination of definitions and assumptions; we have logical axioms which are self consistent within the system of logic they define, while we have non-logical axioms which define things within the system of logic, but are taken as true without proof. We take the mathematical axioms to be the groundwork for any reasoning, which we only call valid if it does not lead to contradiction.

Now with this groundwork we can create hypothetical premises possible within this system of logic and see if they lead to contradiction (and are therefore invalid), or they can be consistent with the system of logic and then are valid. Our choice of axioms can lead to either meaningful statements (as is the goal of mathematics), or can lead to meaningless statements.

When considering this we can then say: yes, math is based on assumptions. This however is hardly disputed, a more interesting thing to say is: math is as based on assumptions because everything is but math is more general, more abstract, and more axiomatic than most fields of study.

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If we can't ontologically prove the existence of reality then ontology is redundant. We need another way to proceed as it seems reasonable there is a reality with well demonstrated evidence that it is there. You say we have to assume the existence of reality before we can develop ideas in reality. Yes, but proceeding from assumption to demonstration of an idea is not what makes our ideas work in reality. Reality works that way. We merely discovered it. No matter how abstract, thinking has not been demonstrated to take place outside a "real" human brain and it is in that sense real. –  Geoff Pointer Nov 7 '13 at 0:32
    
yes, I agree. The point, however., is that if math works in reality and we don't assume that reality is there then it follows that math (within its own system of logic) is not necessarily right. –  Juan Sebastian Lozano Muñoz Nov 9 '13 at 4:55
    
That makes no sense to me. It's not just about the act of assuming. We can assume all kinds of things that amount to nothing. Assuming something that turns out to be correct does not make the assumption the point. The Sydney harbour bridge is not built out of assumptions, it's built out of the real stuff that we assumed correctly about. The original statement is about maths being built on assumptions. Well it's not just assumptions, but assumptions that turned out to be correct. Once they're demonstrated to be correct, they're not just assumptions any more. –  Geoff Pointer Nov 12 '13 at 8:32
    
I'm not talking about axioms or other things (i.e definitions) that make the math itself work, I'm talking about the assumptions about things that can't be demonstrated to be real (metaphysical idealism, for example, can't be dis-proven, but it can be shown to be unreasonable) but are there because without them the math would be meaningless. I don't think we are making different points, but approaching the same point from different perspectives: we make the assumptions that are reasonable, because they have been shown to be reliable and meaningful. –  Juan Sebastian Lozano Muñoz Nov 12 '13 at 11:29
    
Cont. "proceeding from assumption to demonstration of an idea is not what makes our ideas work in reality" (from your first comment) is exactly my point, we assume reality because math works within reality, and math can be deemed reasonable. The statement "If cats live on earth they breath air" assumes that cats can exist, which is not fundamental to the point of the argument (reality existing is hardly relevant to the proof of Fermat's last theorem), but is fundamental to the meaningfulness of the argument (if reality didn't exist then any proof within reality would be meaningless to us). –  Juan Sebastian Lozano Muñoz Nov 12 '13 at 11:39

HINT: Assume everything isn't built on assumptions. This is a contradiction. Thus everything must be built on assumptions.

(Why the down votes?)

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Are we "assuming" that everything is not built on assumptions –  Anz Joy Nov 5 '13 at 16:51
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Yes, which is a contradiction. –  Don Larynx Nov 5 '13 at 16:57
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Thus not everything isn't built on assumptions. What about math though? –  anon Nov 5 '13 at 17:01
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This answer, while amusing, is conflating metalanguage and some fixed set of mathematics. –  Jason Polak Nov 5 '13 at 17:05
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That's roughly the point of that statement in the first place. I came up with that years ago when arguing with the kind of philosophy that supposedly demonstrates we can't have absolute truth and then goes on to assume that everything is purely down to individual observation. If there are no absolute truths then that statement is an absolute truth, which is a contradiction and so those that claim there are no absolute truths using that kind of sophistry are contradicting themselves. On the other hand, if there are absolute truths then that claim is just simply false. So lets move on. –  Geoff Pointer Nov 17 '13 at 2:44

Mathematical and ordinary language and reasoning are the same in this respect.

If there is a general question of whether and how human thought relies on hypotheticals, that's an interesting subject to explore, but the only reason it appears to be happening more often in mathematics is that the rules of mathematical speech are more exacting and force the assumptions to be brought out explicitly. In everyday communication there is a larger collection of assumptions, most of which are not articulated, or are assumed (often wrongly) to be carried by context or shared background.

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On a lighter note:

A maths teacher in the UK began a solution to a problem with the words "Suppose $x$ is the number of sheep that solves the problem ..."

only to be asked by a student at the front of the class:

"what if $x$ isn't the number of sheep ...?"

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Numbers are assumptions, check the axioms which Giuseppe Peano wrote. And more interestingly, these assumptions are not enough to proof what mathematics can express (see Gödel's incompleteness theorems)

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Peano wrote some assumptions (axioms) about numbers, but this doesn't mean that numbers themselves are axioms. –  Trevor Wilson Nov 5 '13 at 21:58
    
@TrevorWilson: True, however, natural numbers do not necessarily correspond to any physical phenomenon, so I would call them axioms. (In fact they do, as do primes, but that’s beside the point.) –  Jon Purdy Nov 6 '13 at 8:39
    
@TrevorWilson well numbers are not axioms but constructs based on them (I'm not talking about "quatities" but "numbers"). Before Peano numbers where based on "instinct" but that's not enough for any scientific proof. –  estani Nov 6 '13 at 9:35
    
I didn't mean to make a philosophical point about whether or not numbers actually exist, only to point out that the word "assumption" usually means something we consider as true (rather than false) and that "truth" and "falsity" are more aptly applied to propositions about numbers rather than to numbers themselves. –  Trevor Wilson Nov 6 '13 at 18:13

In a sense, one can also argue that at a fundamental level, mathematics presupposes the correctness of the rules of logical reasoning.

That's why axioms are taken to be self-evident; they are only self-evident if you assume the correctness of the rules of logical reasoning.

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How do the axioms (for example ZF(C)) become self-evident from assuming the correctness of the rules of logical reasoning? –  Tobias Kildetoft Nov 5 '13 at 20:27
    
As I'm sure you know, ZFC is not "self-evident" (in fact, the history of set theory proves how not-self-evident they are), and are certainly not deducible from the first order logic. :) –  nomen Nov 6 '13 at 8:45

Mathematics describes objects in a formal way. All objects proceed from some base assumptions which are called axioms. The entire logic of "doing" mathematics can be summarized as:

If we have a system consisting of these axioms, then various interesting things emerge.

Sometimes it happens that the axioms resemble something in the real world: perhaps in nature, or in a man-made system like finance and whatnot, and then those interesting results give us rich models which make useful predictions: math and logic find an application.

You cannot disagree with axioms; they have to be accepted. To disagree with an axiom is basically to say "I do not like this system; I want to discuss another system". Firstly, that other system will also have its own axioms, so you cannot escape from axioms. Secondly, you cannot deny the existence of an object by denying its axioms: this is similar to the way the world does not go away when you shut your eyes and plug your ears.

Any set of axioms that do not contradict each other "exist" in a sense. A system of axioms A, B, C exists. And so does a system of axioms not A, B, C: one in which B and C hold, but A is a falsehood. For instance see Euclid's fifth postulate.

Sometimes the word assume is used very loosely. It is legitimate for us to invent a variable $x$ and give it a value. Whether we express this as "let us assume that $x$ is 0" or "let $x$ be 0" is just a matter of wording.

That $\sqrt-1 = i$ isn't an assumption; rather $i$ is just the name that we give to a special number which has the property that $i\cdot i = -1$. If we postulate it as an axiom that such a number exists, and incorporate it into algebra, then that leads to certain results and those results are consistent and useful, giving us the complex numbers and all that follows from them. It is the consistency which makes $i$ legitimate.

If we postulate that $\frac{1}{\infty} = 0$, that will lead to inconsistencies, by means of which we can prove the same statement to be either true or false based on which steps we take. $\frac{1}{\infty} = 0$ is bunk.

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Your examples shouldn't properly be called assumptions. They are actually (chronologically): a premise, a (not generally accepted) axiom and a definition.

Concepts such as 'assumption', 'supposition', 'precondition', 'premise', 'axiom' and the like are easily confused with each other. What they have in common is that they tell you something is to be taken to be true. As a philosopher I would say the term 'assumption' can be said to be the broader one, which covers all others. The difference can be seen by looking at when an assumption is to be retracted. A supposition might be retracted within an argument, giving rise to a conditional statement. A premise, on the other hand, is generally retracted after the argument - after reasoning with the casus at hand. An axiom may be unretracted after an argument - covering multiple arguments. (Note that I am using the word 'argument' in the sense similar to how it is used in "I had an argument with my dad") A lot more can be said on this subject, from a philosophical view.

In general the difference is to be explicated by philosophy, but in the case of mathematics, some strict rules have been formed. We can subdivide the concepts into three categories.

  • $A$ is a supposition when assuming $A$ we can derive $B$ in order to conclude $A \to B$
  • $A$ is a premise when assuming $A$ we can derive $B$ in order to conclude $A \vdash B$
  • $A$ is an axiom when we assume $A$ otherwise. (There is no axiom-specific syntax.)

To say that we assume so much because math is the analysis of hypotheticals could effectively be seen as conflating suppositions with other forms of assumptions.

To answer the question why are there so many assumptions in mathematics, I'd like to refer to Asaf Karagila:

Mathematics is a science of deductions.

Such a science is useful when we in fact do hold something to be true, to which a mathematical deduction is applicable. The following sentence is true, uses mathematics, and is void of any (mathematical) assumption (except maybe for axioms): "I am holding one pen in my left hand and two pens in my right, so I am holding three pens."

The relation between axioms and definitions is a difficult one (for example: don't axioms define what we mean by addition?), which I will not get into right now.

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There are a couple of situations where you come across the phrase assume B, among which are

In neither case assumptions as in the every-day meaning of the word are made. The word assume rather takes the role of a marker which most of the time stands right before

  • the (false) statement which is disproven by reductio ad absurdum

    Assume $\sqrt{2}$ is rational, then...

  • the specification of a case in a proof by exhaustion

    Case 1: Assume $x<0$

  • $A$ in a proof of $A \Rightarrow B$ (although the marker Let is more common)

  • the induction hypothesis

    assume $P$ holds for some $n \in \mathbb{N}$, then...

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The very Logic we build Mathematics from is itself built from assumptions; for example, there is the assumption that it is free from contradictions. See Priest's An Introduction to Non-Classical Logic: From If to Is.

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Why the downvotes? –  Shaun Nov 26 '13 at 17:23

Maths gives us the liberty of choosing our axioms and the rule of inference so long as they do not lead to a contradiction. That said, most of the familiar mathematics is based on first order predicate logic and the existence of at least one infinite set, namely the set of Natural Numbers.The assumption that one can "bundle" together an infinite number of propositions into one single finite universal sentence seems to me to be a critical one. Without that, we cannot even say ' every natural number has a successor'.

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protected by Michael Greinecker Nov 6 '13 at 0:30

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