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:
So would you guys mind telling whether math is built on assumptions?
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.
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.
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.
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.
$\frac{1}{\infty} = 0$ 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.
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.
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.
I think there are a few interpretation of the question "why we assume so much in math". I would answer some of them follows:
I hope this helps $\ddot\smile$
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?
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.
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.
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.
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 ...?"
HINT: Assume everything isn't built on assumptions. This is a contradiction. Thus everything must be built on assumptions.
(Why the down votes?)
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.
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
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)
assume $P$ holds for some $n \in \mathbb{N}$, then...
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. Minkowski, using some abstract mathematics relating to spaces with unusual metric properties, clarified further the solution to the hugely significant real world problem of special relativity that Einstein had introduced.
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.
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.
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.
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.
About a century and a half ago there was a fairly broad agreement among mathematicians about what constitutes mathematics and it was possible to argue with a straight face that mathematics is not based on assumptions but rather constitutes a kind of certain knowledge that lacks in other fields of human endeavor. The situation changed with the emergence of the foundational challenge by Brouwer. Ever since the beginning of the 20th century there is no universal agreement among mathematicians as to what constitutes a collection of proper rules that would govern such a body of certain knowledge, and therefore it has become more difficult to keep a straight face when making the corresponding claim. Since the emergence of category theory it has become difficult even to maintain that a majority of mathematicians agree as to what constitutes a collection of proper procedures to follow. Even in the context of set theory, alternative approaches have been developed such as Internal Set Theory that some feel is preferable to ZFC as a foundation for mathematics.
So I would say that today there is a rather broad agreement that mathematics is indeed based on assumptions, the most basic of which is the choice of which foundational system to work in.
If you go back in past humans first learnt counting things like one"(|) stone two(||) stones three stones(|||) and so on. This made them understand how they can add two(||) stone and two stone (||) to get four(||||){one can argue first base humans used were in 5) this taught them anti addition or subtraction. Repeated addition taught them multiplication where repeated subtraction lead to division .Which further lead to repeated multiplication or division to exponential making them ask anti exponential or logarithmic function. ANd this is how algebra evolved.SO we can say so far that anything mathematical operation which can be defined with help of natural processes is not assumed. But what about irrational number ?(rational decimal are relatable to natural processes)Most of them I guess origined from Geometry but antiexponential or logarithm also give irrational number! SO atleast we can affirmatively any mathematical operation which can be defined using physical things is not assumed.
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)
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.
