Should I trust mathematics?

Question

First of all I'm not an expert in this field, please correct me if I'm lacking relevant knowledge here. A few hundreds years ago mathematics was largery based on intuition. People realised we need to make mathematics more rigorous. Axioms and definitions came into play and people have chosen to derive new theorems from those axioms and definitions by using propositional logic. But the axioms and definitions were formulated by mathematicians to reflect some informal ideas, such as a definite integral, or some much more abstract like those in topology. How are mathematical definitons formulated? We start with a specific example, find some more examples and notice patterns, features these examples have in common. Let's take definite integrals as an example. It's a result of our attempts to find the area under the graph of a function. What if I formulated the Riemann integral to be the limit of sum of infinitely many infinitely thin rectangles having height f(x)? Today we know that the existence of limit is not enough here, we need to make sure the choice of sample points doesn't affect the value of limit.

So basically I don't know if a new definition is correct until I find a counterexample. A definition has to be simple, capture my intuitive idea of the area under the curve, but not too narrow (I have to impose certain restrictions, i.e. the function has to be integrable) and not too broad (I don't want to define a definite integral so that in special cases it cannot be considered as the area under the curve).

If you look at topology in mathematics, it's full of some crazy definitions that seem to have come from nowhere. Indeed, it took mathematicians a few decades in 20th century before the majority accepted this set of definitions. If it took so long to formulate them, then the earlier definitions had to be "worse" in some sense. Does it mean the current ones are perfect? For some reason, probably not. Maybe it was more like: here is our (mathematicians of 20th century) idea of what topology should be all about, and finally here's the set of definitions that fullfills these needs. But we cannot be sure it does - we can find a counterexample showing that those definitions are not "complete" in some way, leading to serious paradoxes etc.

So my question is - how to make sure a definition captures the intutive reasoning correctly, taking all special cases into account? I'm guessing the answer is: you can't do that, because definitions are axioms. But remember, we cannot do maths without the use of intuition, no matter how rigorous we are trying to be. Even Russell's Principia Mathematica assumes the reader will interpret the magical symbols in the same way as the author. The assumption that we are all thinking in the same way "on a certain level" - let's allows us to do mathematics. Imagine a small child learning how to speak - does he know any language he can refer to in order to learn how to speak? Obviously not. We learned human speech as kids from specific examples of how and when certain sentences and words are used, in which situations. We can communicate with each other because the way we interpreted these examples was identical on some level.

The problem I'm trying to emphasise here is that we actually can develop theorems from certain definitions (axioms) for years and then realise the axioms were wrong in some way, incorrect with our intentions. Then it'd be practically impossible to "repair" this theory. Mathematics is correct until we find a flaw in it.

---

UPDATE

One more, important example:

Continuous function. In the times of Newton and Leibniz, mathematics was largery based on intuition. They developed a theory without being precise in what limit or continuity meant, they didn't have a formal definition of these notions. The intuitve motivation for the definition of a continuous function is the ability to draw a graph of it without lifting the pencil from a plane. This isn't rigorous. The aim was to find a formal definition. It was first formulated as: function f is continuous at point x if:

No matter how hard I try, I can't find an example that this definition doesn't agree with this intuitive notion of continuity. Still, there is a chance someone smarter will find it. Why do we confirm such a definition by giving examples showing that it works? We can never prove anything by examples (we can only disprove a theorem by providing a counterexample)! Somehow all mathematicians agree this definition is correct. What makes them think so?

Answer 1

... how to make sure a definition captures the intutive reasoning correctly, taking all special cases into account?

Well that's the art! In his preface to Calculus on Manifolds, Michael Spivak says:

There are good reasons why the theorems should all be easy and the definitions hard.

That quote says it all. The art is to find the right definitions. You mentioned topology. As you know, it took decades of struggle to find the right characterization of continuity. The realization that the notion of the open set was the essence of continuity was a huge breakthrough. It's only obvious in retrospect.

You mentioned Riemann integration. That's another good example, because in higher math the Riemann integral is no longer used. There's a more general theory called Lebesgue integration which behaves better. That doesn't mean that we were wrong to "trust" Riemann. Trust really has nothing to do with it. We live in the world as it is.

The struggle to find good definitions is at the core of the development of mathematics. So when you ask, "... how to make sure a definition captures the intutive reasoning correctly, taking all special cases into account? the answer is ... if we knew, we'd bottle it and give it to the undergrads. There's no magic formula for progress.

But you are also asking if we can "trust" experts, knowing that in a few decades they'll be proved wrong or foolish. If you had an infected leg in the 1800's they sawed it off. Without anesthetic. Did you trust your doctor back then? What other choice did you have?

It's the same in every field. Math is no different. Mathematics is a historically contingent human activity. It's never perfect but it's always getting better. Painstaking struggle, false starts, the occasional genius, lots of plain old hard work. That's how progress is made in every field.

Whether you trust, and what you trust, is up to you. You drive over bridges. Sometimes the bridges fall down. Over the years we learn to make better bridges, never perfect bridges. You'd be foolish to trust all the experts all the time. But you'd be even more foolish to never leave the house for fear of a falling bridge. Someone the other day asked the difference between rationality and logic. Rationality is what lets you drive over a bridge that you know might fall down, even though you can never personally investigate every nut, bolt, and corrupt government contract.

I just happened to run across a guy named Ignaz Semmelweis. He was a German doctor in the 1840's who said that obstetric deaths could be reduced if doctors would just wash their hands before delivering babies. He was rejected by the medical community, committed to an insane asylum, and beaten to death by the guards.

That's human progress.

Answer 2

The exclusion of pathological examples (i.e. the property of being "not too broad") is not the only desideratum in a definition. You also want the definition to allow certain natural constructions, so that, for example, you can construct products, quotients, coproducts, etc. Sometimes, in order for these constructions to make sense, you have to allow pathological examples. (For example, if you know some algebraic geometry, think about the quotient of a scheme by a subscheme).

In other words, if you insist on having good objects, you might get a bad category of objects, or vice versa. One reason to tinker with definitions is that it forces us to explore this tradeoff.

Answer 3

As a mathematics student I had a teacher who said definitions are never wrong. When you define something, is defined! I like to think mathematics as a game. When you create a game, you define rules. Eventualy if those rules aren't clear enough, players may start do some stuff you didn't tought they would do. But a rule isn't either correct or incorrect...is just a rule! If you don't follow that rule, you are incorrect. If you don't trust a rule... well that just makes no sense does it? Your questions seems more apropriated to physics than mathematics! Think mathematics as chess. Would you even question if you trust the rules of chess? Once the rules are established what players(mathematicians) do is to see what is possible to do with those rules. Maybe I didn't understood your question well enough... or maybe you were talking more about the applied part of mathematics, where people try to make definitions to try explain what they see happening like mechanics. The truth is the further you go in mathematics, more abstract things get! You eventualy hit a point where your intuition isn't usefull anymore. At that point, how would you even feel like a definition seems correct or not?

Answer 4

Given that user4894 already answered your question let me answer the Update.

In fact, I believe that answering the Update is going to be very illuminating to the core of your question, and the primary reason for that is:

The formal definition of continuous function that you presented is incorrect!

Let's see why. Your definition, if stated in more human-readable form:

Function f is continuous iff for any given (small) number epsilon you can find the corresponding number delta, so that for any number y if x differs from y by at most delta the corresponding values of f differ by at most epsilon. That is, small changes in the function argument cause only small changes in the function value.

Sounds reasonable, right? Now try answer the following question:

According to your definition, is function f(x)=1/x continuous on open interval (0, infinity)?

Spoiler alert:

The answer is NO. Even though f(x)=1/x is (according to the standard definition) continuous on the open interval (0, infinity) it is not according to your definition.

To see why take epsilon=1 (any positive number would do), and for any number delta that one would propose pick x and y in the interval (0, delta), but with x much closer to 0 than y. For example, for delta = 1/1000 pick x=1/1000000 and y=1/10000. Because both x and y are in (0, delta) |x-y| < delta will be satisfied. But because x can be picked arbitrary close to 0 their reciprocals can be made arbitrary large; in this example |f(x)-f(y)|=1000000-10000=990000.

Thus your definition does not capture the desired definition of continuity of functions (actually, it captures a somewhat different property, known as "uniform continuity").

To make it right you need to flip the choice of y and the choice of delta:

Function f is continuous iff for any given (small) number epsilon and for any number y you can find the corresponding number delta, such that if x differs from y by at most delta the corresponding values of f differ by at most epsilon. That is, small changes in the function argument cause only small changes in the function value.

Under that definition you look for delta after the number y is fixed. And, under the corrected definition f(x)=1/x is continuous on (0, infinity).

The above example shows exactly why so much care is put into making definition extremely precise. And that bridges to your main question: yes, you can trust mathematics if it's applied correctly, but beware that it may produce unreasonable results if it's applied incorrectly.

Answer 5

If you look at topology in mathematics, it's full of some crazy definitions that seem to have come from nowhere. Indeed, it took mathematicians a few decades in 20th century before the majority accepted this set of definitions. If it took so long to formulate them, then the earlier definitions had to be "worse" in some sense. Does it mean the current ones are perfect? For some reason, probably not. Maybe it was more like: here is our (mathematicians of 20th century) idea of what topology should be all about, and finally here's the set of definitions that fullfills these needs. But we cannot be sure it does - we can find a counterexample showing that those definitions are not "complete" in some way, leading to serious paradoxes etc.

My understanding of the hsitory is that mathematicians proposed definitions and theorems. Those were criticised in various ways, e.g. - the definition included something that should not be covered or excluded something that should be covered. See "Proofs and Refutations" by Lakatos.

So my question is - how to make sure a definition captures the intutive reasoning correctly, taking all special cases into account? I'm guessing the answer is: you can't do that, because definitions are axioms. But remember, we cannot do maths without the use of intuition, no matter how rigorous we are trying to be. Even Russell's Principia Mathematica assumes the reader will interpret the magical symbols in the same way as the author. The assumption that we are all thinking in the same way "on a certain level" - let's allows us to do mathematics. Imagine a small child learning how to speak - does he know any language he can refer to in order to learn how to speak? Obviously not. We learned human speech as kids from specific examples of how and when certain sentences and words are used, in which situations. We can communicate with each other because the way we interpreted these examples was identical on some level.

The problem I'm trying to emphasise here is that we actually can develop theorems from certain definitions (axioms) for years and then realise the axioms were wrong in some way, incorrect with our intentions. Then it'd be practically impossible to "repair" this theory. Mathematics is correct until we find a flaw in it.

All knowledge is discovered by conjecture and criticism. You notice a problem, guess solutions to it, criticise solutions until only one is left and it has no outstanding criticisms, and then you move on to another problem, see "Realism and the Aim of Science" by Karl Popper, chapter I. Maths is no different. Proofs do not prove stuff since their assumptions can be wrong, as you have noted. Rather, they provide interesting target practice. They expose ideas about relationships between different ideas to criticism by making them as precise as you know how to make them. Since they are more precise, more can go wrong with them and there is more you can potentially learn by trying to refute them. Sometimes when a mathematical guess is wrong there is some minor modification that fixes it and sometimes there is no such modification. Some ways of doing this are described in "Realism and the Aim of Science" by Karl Popper, Chapter III (this is about maths although the title is "Metaphysics: Sense or Nonsense?", metaphysics by Popper's lights is just knowledge that isn't experimentally testable and that includes maths).

the relationship between maths and physics is also interesting in this context. The mathematical operations that it is possible for us to think about are those instantiated in the laws of physics. There may be others, but no physical object, including your brain, can model them. So to discover mathematical objects you conjecture that their properties are modelled by the properties of objects you can measure, like marks on a page or data in a computer. It can turn out that those physical objects don't model what you thought they modelled. The way to discover this is to try to model the same mathematical operation in different ways and see if you get different answers to the same question. See "The Fabric of Reality" by David Deutsch, Chapter 10. But it can also turn out that your ideas about the abstraction your are trying to model are false. If you have results that for which you can't find any other explanation, then you may be in that situation.

Answer 6

Study Proof Theory and Model Theory

Proof Theory is about proving syntactic correctness. This is what we often think about when we look at a mathematical proof. It says nothing about whether the result is intuitive.

Model theory is all about proving semantic correctness. This is most likely what you refer to when you talk of an "intuitive" correctness.

There is no straight forward answer to your question, besides perhaps that a careful reading of Proof Theory and Model Theory literature may make you question what you define as "trusting" mathematics. I would argue you should have to define that word before you can give the question a yes or no answer. You will find it hard to do so without a language. That language will really be the deciding factor as to whether you can "trust" math or not.

As a parting thought, here's some thoughts that have been syntactically valid, but semantically troublesome:

• The Greeks were not fans of irrational numbers. As legend would have it, the first guy to prove their existence was thrown overboard for heresy. (we generally now accept that pi is a useful thing)
• Imaginary numbers are... well... imaginary. Their semantic meaning has long been called into question (although generally we now accept that they have meaning, at the very least through Euler's formula)
• In set theory, the axiom of choice seems semantically valid. However, the application of it over infinite sets causes so much unintuitive behavior that many mathematicians reject it.

Answer 7

So basically I don't know if a new definition is correct until I find a counterexample.

You cannot prove that a formal definition is correct, but you can justify it. We can, for example, formally define the natural numbers in terms of a binary function S on a set N as follows:

1. 0 is an element of N
2. For all elements x of N, there exists a unique S(x) also in N.
3. For all elements x and y of N, if S(x)=S(y), then x=y.
4. For all elements x of N, S(x) =/= 0.
5. For all subsets P of N, if 0 is an element of P, and if, for all elements k of P, we also have S(k) in P, then P = N.

How can we justify this definition? First, we can formally prove that (as WillO points out in comments) two systems that satisfy these axioms are essentially identical. Note that a definition need not necessarily define a unique structure, e.g. the definition of a group in abstract algebra.

We can also verify that each of these axioms is consistent with our intuition by various informal arguments.

Then we can derive various theorems from these axioms that we would expect to be true, e.g. that no number is its own successor. The more such theorems we can pile up, the more confident we can feel that the definition is justified.

We can also show that structures such as the set of natural numbers can be derived from other simpler structures. If, for example, we have a set X on which a function f: X --> X is defined such that f is injective but not surjective, the we can extract a subset Y of X such that it is identical in structure in the natural numbers with f as the successor function on Y.

All of these things would make us more confident that the above definition is justified as The Definition of the natural numbers.

EDIT:

What might a "bad" definition of the natural numbers look like? Suppose, for example, we had left out (4) above. Then the remaining axioms would be satisfied by N = {0} and S(0) = 0. This would not be consistent with our intuition of numbers. The axioms should not describe any structure that is not identical to that of the natural numbers. Even if we had not discovered this flaw in our reduced list of axioms, however, we would not have been able to prove various other fundamental properties, e.g. as above, that no number is it's own successor since S(0)=0 would be a distinct possibility. Also, we would not be able to prove, as WillO suggested, that the structure described is uniquely determined by the axioms. There would infinite different possibilities if we left out (4), including, it should be pointed out, the natural numbers themselves.

Answer 8

No matter how hard I try, I can't find an example that this definition doesn't agree with this intuitive notion of continuity.

Maybe you need an example of a discontinuous function.

Consider the step function: f: R --> R where f(x)=0 for x<0, 1 otherwise.

f is discontinuous at x=0 because, given epsilon = 1/2, it is impossible to find a delta such that if |y - 0| < delta then we must have |f(y) - f(0)| < 1/2.

Intuitively, a very small change in x does no always result in very small change in f(x). Going from x = -0.000 000 000 000 1 to x=0, for example, results in an relatively huge increase of a whole unit. Of course, this begs the question, what is "small." That's why we need a formal definition.

Somehow all mathematicians agree this definition is correct. What makes them think so?

It works. It fits our intuition of what both continuous and discontinuous functions ought to be like.

Answer 9

Mathematics is the study of alleatory universes. I.e., let U be a universe with A1, A2, A3. (...) One of them is "our" universe. Physical universe.