8

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:

enter image description here

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?

  • 5
    Math is a historically contingent human activity. You can trust it as well as you can trust anything made by people. Which is to say, only up a a point. You ride in cars, right? They're about as reliable as math. But surely you don't doubt that 2 + 2 = 4. What is your actual concern here? That someone will realize we're using the wrong definition for simplicially enriched categories and your world will come tumbling down? What's the real concern here? en.wikipedia.org/wiki/… – user4894 Feb 16 '15 at 20:20
  • 5
    Axioms don't have to be correct, just consistent; mathematicians have on occasion purposefully used axioms that do not apply to the real world in order to investigate problems that are otherwise intractable. – Matthew Read Feb 17 '15 at 2:34
  • @MatthewRead, one could as well say that axioms don't have to be consistent, just correct. You have a "meta-axiom" in play here defining that axioms which are not consistent (defined by another meta-axiom) are not acceptable to you. This is an interesting philosophy, but it's your philosophy, and is not inseparable from mathematics as you seem to believe. If you grade knowledge according to accuracy of predicted observations and not as a binary state (true or false), you get another philosophy entirely, and one that is fully compatible with mathematics AND physics. – Wildcard Aug 29 '17 at 0:41
10

... 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.

  • I think the comparison of mathematical progress to medical progress, bridges, etc. is lacking. The way that building a bridge can go wrong is radically different from a mathematical definition going wrong. In the first case, you rely on something which doesn't work, and in the second case, everything you rely on does work, it just doesn't quite capture what you might want it to capture. – 6005 Feb 17 '15 at 23:51
  • @Goos There are thousand year old wooden bridges still in use in China. The materials have been replaced over the years but the original design and engineering persist. Bridges can be as eternal as math. I hold math as a historically contingent human activity. You raise a good point and I need to think about it more. But math is not the only discipline with eternal truths. Medicine is about healing, for example. That never changes. – user4894 Feb 18 '15 at 0:41
  • That sounds reasonable. How would you comment the example of continuous functions I've described in the question update? – user107986 Feb 19 '15 at 15:02
  • @user107986 You should post a new question. It's hard to respond to this in only a comment. FWIW that's not the most general definition of continuity. In topology, the definition is "The inverse image of an open set is open." That lets you define continuity in settings where there is no notion of distance to do epsilons and deltas with. It's a nice exercise to prove that this definition generalizes the epsilon/delta definition. But I don't quite see your point except by stating the obvious. The def you gave characterizes continuity in metric spaces very nicely. 240 yrs from Newton to Zermelo. – user4894 Feb 19 '15 at 17:22
  • 2
    @user107986 I can't see getting into a lengthy dialog by continually altering an answer people have already read and voted on. I'm not a Stackexchange expert but I'm pretty sure that's counter to the spirit of things around here. But the topological definition of continuity does not "falsify" the epsilon def; it generalizes it. And in contemporary math people are still struggling to find the right definitions. n-categories are a contemporary example I'm aware of. Each author uses different definitions because the ideas are being worked out. What of it? We're in violent agreement on this point. – user4894 Feb 19 '15 at 18:03
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.

  • "if you insist on having good objects, you might get a bad category of objects" I would be very interested if you would elaborate on that comment with some specific mathematical examples. That would prove very enlightening. Especially if you have examples at the level of groups, rings, top spaces, etc. that don't involve too much (ie any) algebraic geometry. – user4894 Feb 17 '15 at 0:17
  • @user4894, the classic example of this principle at work is studying commutative unital rings rather than the less general integral domains. (While interesting, I'm not sure this truly helps answer the question.) – goblin Feb 18 '15 at 7:28
1

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?

  • Eventually you reach a point where your intuition isn't useful anymore I don't think this is necessarily true. Even more abstract mathematics can have an intuition around it, it just requires a depth of understanding to make the relevant connections. This intuition may not be guided by a concrete example directly applicable to the real world, but there certainly can be some type of intuition. – Graham Sep 20 '18 at 3:27
1

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.

  • Congrats on evidently being the only person who carefully read the OP's definition. Good catch. – user4894 Feb 26 '15 at 0:16
  • I've taken it from here (page 1), how can it be incorrect. It's just Cacuhy's definition. – user107986 Feb 26 '15 at 7:33
  • 1
    Well, the original Cauchy's definition is now referred to as "uniform continuity". :) It's equivalent to "continuity" for compact domains, but, as the 1/x example shows, it's slightly too restrictive for non-compact domains. – Michael Feb 26 '15 at 7:45
  • @user107986, a bit of clarification: currently continuous functions are understood as those that satisfy Weierstrass's definition. – Michael Feb 26 '15 at 7:55
  • Ok, fine. Why did you say that my definition is incorrect? There are no incorrect definitions. Did you mean it doesn't agree with our intuitive notion of continuity, i.e. you can easily give an example of a function that's considered continuous, but is not continuous according to this definition? If so, we don't know if the current definition is correct (until we find a counterexample). – user107986 Feb 26 '15 at 13:43
0

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.
  • You might consider the highly counterintuitive behavior that the negation of AC causes. A vector space with no basis. An infinite set not cardinally equivalent to its square. The loss of the trichotomy law for infinite cardinals. It's arguable that the negation of Choice is far more intuitively unappealing than the acceptance of Choice. People never think about this but there's a good reason mathematicians accept Choice. Can you name any mathematicians who "reject" Choice who are not already constructivists and reject many non-Choice infinitary constructions? – user4894 Feb 16 '15 at 22:08
  • 1
    It's definitely a rare mathematician who rejects the axiom of choice. It's apparently a common misconception that such a thing is considered normal. – Matt Samuel Feb 17 '15 at 19:19
0

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.

  • I don't know why this answer was downvoted. "Proofs do not prove stuff since their assumptions can be wrong, as you have noted." Yep, that is exactly right. If I prove a theorem T using a formal system F, then I have not proved T per se, I have only proved that F proves T. – goblin Feb 18 '15 at 7:37
0

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.

  • For sake of discussion ... the epsilon/delta definition falsifies the heck out of our intuition. I put forth for your consideration the Weirstrass function, which is continuous at every point and differentiable at no point. The brilliance of a great definition is that it challenges and extends our intuition. – user4894 Feb 19 '15 at 19:40
  • @Dan Christensen, you've probably misunderstood my point. I mean this definition is most likely correct. No matter how hard I try, I can't find an example that this definition doesn't agree with this intuitive notion of continuity. - in other words, all continuous I can think of satisfy conditions in the formal definition. Similarly, all discontinuous functions I can think of do not satisfy the se conditions. The main question here is - what made Cauchy believe this is a good definition of continuous function? – user107986 Feb 19 '15 at 20:00
  • @user107986 As a concocter of a few definitions myself, my guess is that Cauchy probably went through of number of possibilities, rejecting them because they did not work in some cases or where just plain unworkable, i.e. too difficult to use in applications. Writing a definition in mathematics is a bit like writing a computer program. It has to be thoroughly tested to get any "bugs" out. – Dan Christensen Feb 19 '15 at 20:43
  • But, as every computer program, no matter how much time you spent on it, someone will always find holes or bugs. – user107986 Feb 19 '15 at 21:10
  • So, you don't trust computers? Throw yours in the trash and move into a cave. ;^) – Dan Christensen Feb 19 '15 at 22:01
-1

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.

  • 1
    This is not a definition of the natural numbers until you prove that there is a unique structure satisfying your axioms. – WillO Feb 17 '15 at 0:05
  • @WillO Good point. I should have mentioned that. It is easy to do though. See my posting, "Daddy, where do numbers come from?" (Feb. 19, 2013) at my math blog dcproof.wordpress.com – Dan Christensen Feb 17 '15 at 4:47
  • 1
    -1, I'm tired of people misusing the word "definition." – goblin Feb 18 '15 at 7:39
  • I'm tired of people who think their own narrow, technical understanding of a word is the only one permitted. In any case, "definition", although very useful, is an entirely informal concept in mathematics. There is no formal notion of a "definition" in FOL, for example. – Dan Christensen Feb 18 '15 at 14:22
-2

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.

  • 6
    Which version of mathematics is actually "our" universe? I'd argue none of them. We know of no continuous quantities, no infinite aggregates. Math can be imperfectly applied to certain aspect of reality; but there is no math system I know of that represents our physical world. Happy to take any corrections you can supply. – user4894 Feb 16 '15 at 20:03
  • @user4894 Here is a correction: You, as anyone else, do not know precisely how the physical universe works. You are not qualified to assert that mathematics cannot represent it. – Matt Samuel Feb 19 '15 at 2:17
  • @MattSamuel I had hoped my phrases "We know of no ..." ... and "no math system I know of ..." would communicate my awareness of the limits of my knowledge. I was attempting to refute Vinicius, who claimed that there is some mathematical system that fully represents our world. IMO it's that claim that has the burden of proof. Did you feel that I made an absolute assertion? I shall try to write more clearly in the future. – user4894 Feb 19 '15 at 3:11
  • Let $F$ be a set of axioms. Let $\{F \}$ be the set of possible sets of axioms. There exists one only $F_0$ which coincides with True Reality. – Vinícius Ferraz Jun 5 '15 at 1:43

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.