Math Preference: Inductive vs. Deductive Reasoning

[whyevengothere]

I have an incredible distaste for the axiomatic approach ,it's a very bad method,I think ,for teaching or learning about mathematics.I don't understand why I feel this way, I always thought inductive reasoning in mathematics ,the sort you find with physicist,is better than the deductive reasoning you find with the Bourbaki group.
What do math people on this forum prefer ? Why ? How is it better?
Any comment?
P.S here's a very interesting and somewhat related quote from one of George Polya's books:
''Induction often begins with observation. A naturalist may observe bird life, a crystallographer the shapes of crystals. A mathematician, interested in the Theory of Numbers, observes the
properties of the integers 1, 2, 3, 4, 5, . . . . If you wish to observe bird life with some chance of obtaining interesting results, you should be somewhat familiar with birds, interested in birds perhaps you should even like birds. Similarly, if you wish to observe the numbers, you should be interested in, and somewhat familiar with, them. You should distinguish even and odd numbers, you should know the squares 1,4,9,16,25, . . . and the primes 2,3,5,7, 11, 13, 17, 19,23,
29, . . .. Even with so modest a knowledge you may be able to observe something interesting.''

 

[1MileCrash]

Inductive reasoning has absolutely no place in mathematics in my opinion.

 

[whyevengothere]

Inductive reasoning has absolutely no place in mathematics in my opinion.

Some very good mathematicians like Newton,Euler,Poincaré,Minkowski, Weyl, Kolmogorov were prominent users of it,according to Vladimir Arnold who's also one of them,on other side you may find Leibniz and Descatres,the Bourbakists,Artin,Noether.
I think the best discoveries in math were made this way(induction).

 

[1MileCrash]

Inductive reasoning cannot prove anything by its very nature. I won't argue that it isn't useful for making conjectures, but it can never be used in proof.

 

[edward]

I thought Axiomatics were a hand held calculators sold on late night infomercials.

 

[jostpuur]

A related joke: http://abstrusegoose.com/504

The thread opening reminded me about my attempts to study Galois theory. All books on it are perfect examples of where rigor definitions are merely piled up on each other with no apparent motivation, and they are extremely difficult to understand.

Inductive reasoning has absolutely no place in mathematics in my opinion.

Have you attempted to understand Galois theory? Have you succeeded or failed?

 

[Matterwave]

Inductive reasoning cannot prove anything by its very nature. I won't argue that it isn't useful for making conjectures, but it can never be used in proof.

Are you saying proofs by induction are invalid?

 

[jostpuur]

These are supposed to be different things:

http://en.wikipedia.org/wiki/Inductive_reasoning

http://en.wikipedia.org/wiki/Mathematical_induction

1MileCrash already contradicted himself by first stating

Inductive reasoning has absolutely no place in mathematics in my opinion.

and then

I won't argue that it isn't useful for making conjectures

so obviously he isn't choosing his words carefully.

Anyway, I'm sure we know that whyevengothere is talking about a real thing. Sometimes mathematics is pure axiomatic definitions piled on each other, and it can get incomprehensible.

 

[disregardthat]

The axiomatic approach is entirely necessary, especially in modern mathematics, to even do the slightest actual work of mathematics.

 

[jostpuur]

Sounds simple when your are talking about this on the general level, but has anyone here totally enjoyed studying Galois theory? In Galois theory you are given properties of groups and fields in such way that you have no clue how the pieces are supposed to work in the end. I haven't liked it, I can admit on my part.

 

[micromass]

Some very good mathematicians like Newton,Euler,Poincaré,Minkowski, Weyl, Kolmogorov were prominent users of it,according to Vladimir Arnold who's also one of them

I hope you know that all of these people used axioms and deductive reasoning.

 

[disregardthat]

Sounds simple when your are talking about this on the general level, but has anyone here totally enjoyed studying Galois theory? In Galois theory you are given properties of groups and fields in such way that you have no clue how the pieces are supposed to work in the end. I haven't liked it, I can admit on my part.

Not trying to assume too much, but it seems to me that you are equating the "axiomatic approach" with a way of presenting mathematics for the student, and not a way of presenting mathematics as a subject.

If a book teaches Galois theory intuitively, with lots of examples and outside reasoning for every new piece, that would not undermine what we understand as the "axiomatic approach". I personally found the introduction to galois theory when I first encountered it in the book on Abstract algebra by Fraleigh as very difficult to put together in my mind. I do not however blame the general approach, just that particular presentation or set-up.

 

[microsansfil]

Hello

In my analysis

Inductive reasoning leads from observations ( (and what we imagine to accommodate the observations in a conceptual framework) to the model (Physics).

Once the formal model in hand, we proceed by deduction to make predictions (mathematics, calculus).

Patrick

 

[whyevengothere]

I hope you know that all of these people used axioms and deductive reasoning.

Yes ,maybe, but I think that most of their mathematics was done ''experimentally''( there's a video you can google by V.I Arnold titled ''mathématique expérimentale'' online,you should see it).

 

[whyevengothere]

Hello

In my analysis

Inductive reasoning leads from observations ( (and what we imagine to accommodate the observations in a conceptual framework) to the model (Physics).

Once the formal model in hand, we proceed by deduction to make predictions (mathematics, calculus).

Patrick

Yes ,that's exactly what I meant.

 

[micromass]

Yes ,maybe, but I think that most of their mathematics was done ''experimentally''( there's a video you can google by V.I Arnold titled ''mathématique expérimentale'' online,you should see it).

Yes, of course. Mathematics is an experimental science in the sense that every mathematician first does some experiments on known objects and known theories in order to obtain something new. However, after this, they proceed with proving their theories rigorously using deductive reasoning. This is simply how mathematical research works.

 

[microsansfil]

Yes ,that's exactly what I meant.

A Professor of Mathematics fan of George Polya’s classic Mathematics and Plausible Reasoning.

Patrick

 

[1MileCrash]

Are you saying proofs by induction are invalid?

Proofs by mathematical induction are not inductive reasoning. It's deductive reasoning. If I prove something via mathematical induction, it's over, the theorem is true. Yes, a mathematical "proof" that uses inductive reasoning is not a proof, completely invalid, and means nothing.

A "proof" using inductive reasoning would be something like "well, for the first hundred natural n, the sum from 1 to n is n(n+1)/2. C'mon man, it works for the first hundred, that makes me feel a lot of feelings. Therefore, for all natural numbers, the sum from 1 to n is n(n+1)/2."

These are supposed to be different things:

http://en.wikipedia.org/wiki/Inductive_reasoning

http://en.wikipedia.org/wiki/Mathematical_induction

1MileCrash already contradicted himself by first stating
and then
so obviously he isn't choosing his words carefully.

Anyway, I'm sure we know that whyevengothere is talking about a real thing. Sometimes mathematics is pure axiomatic definitions piled on each other, and it can get incomprehensible.

What I mean is that inductive reasoning can be used for us to come up with "inklings" or conjectures that we think may be true (IE, they give us the idea of a problem to solve), but in actually doing mathematics (ie, proving theorems or solving problems) inductive reasoning has no place. Show me one theorem proven with inductive reasoning, and you win. Inductive reasoning by definition does not guarantee its consequent, it doesn't prove anything, ever.

 

[whyevengothere]

Proofs by mathematical induction are not inductive reasoning. It's deductive reasoning. If I prove something via mathematical induction, it's over, the theorem is true. Yes, a mathematical "proof" that uses inductive reasoning is not a proof, completely invalid, and means nothing.

A "proof" using inductive reasoning would be something like "well, for the first hundred natural n, the sum from 1 to n is n(n+1)/2. C'mon man, it works for the first hundred, that makes me feel a lot of feelings. Therefore, for all natural numbers, the sum from 1 to n is n(n+1)/2."



What I mean is that inductive reasoning can be used for us to come up with "inklings" or conjectures that we think may be true (IE, they give us the idea of a problem to solve), but in actually doing mathematics (ie, proving theorems or solving problems) inductive reasoning has no place. Show me one theorem proven with inductive reasoning, and you win. Inductive reasoning by definition does not guarantee its consequent, it doesn't prove anything, ever.

Read Polya 's book ''Mathematics and Plausible Reasoning''.

 

[WWGD]

I can summarize my perspective: I know "they" (concepts, theorems) are abstractions .
Abstractions of _what_? Information can be lost in the process of abstracting. What was the
author thinking, aiming for when s/he coined the term?

 

[micromass]

Read Polya 's book ''Mathematics and Plausible Reasoning''.

Can you just summarize your point instead of telling us to read an entire book?

 

[whyevengothere]

Deductive reasoning is incapable of yielding essentially new knowledge,plausible reasoning can and does all the time,therefore it should occupy a larger part of the teaching and learning about math,the axiomatic method require that one accepts any axiom with a hope that its corollaries are fruitful,and this just causes me a great discomfort ,all those non-motivated definitions ,ugh...and I'm just asking if this is reasonable or not.

 

[disregardthat]

Deductive reasoning is incapable of yielding essentially new knowledge,plausible reasoning can and does all the time,therefore it should occupy a larger part of the teaching and learning about math,the axiomatic method require that one accepts any axiom with a hope that its corollaries are fruitful,and this just causes me a great discomfort ,all those non-motivated definitions ,ugh...and I'm just asking if this is reasonable or not.

Inductive reasoning is only useful because it leads to deductive reasoning. Deductive reasoning is the goal, the goal is always to find new theorems and results. How you do it, by inductive reasoning or otherwise, is only secondary to this goal. Axioms are, usually by default, heavily motivated. Take any axiom, and read its history, and you will see why it was introduced. Creating new theories does not only require you to prove theorems, but also to abstract, which often means creating new axioms.

Quite contrary to what you're saying, deductive reasoning is actually the only possible method capable of yielding essentially new knowledge.

 

[micromass]

Deductive reasoning is incapable of yielding essentially new knowledge,plausible reasoning can and does all the time,therefore it should occupy a larger part of the teaching and learning about math,

Sure, that is completely true. But deductive reasoning is absolutely essential in checking the results and presenting them. I have plausible ideas all the time, and only 1% of them ever works if I check it rigorously. So axiomatics and deductive reasoning are extremely important in mathematics and are responsible for making math correct and trustworthy.

In teaching of math, both deductive and plausible reasoning should get attention, and both do get attention. Students should absolutely learn about logic and deductive reasoning because they should know how to check that their arguments are correct. It is only a deductive argument that is accepted in math (rightfully so), so the students should learn it. That said, plausible reasoning also occupies a large part of learning mathematics. In fact, whenever you do exercises, the teacher will teach you about plausible reasoning (if he's any good).

the axiomatic method require that one accepts any axiom with a hope that its corollaries are fruitful,

Not at all. Axioms can and should be motivated. Blindly accepted axioms with the hope that something fruitful comes out is the wrong way of doing thing.

That said, when learning math, we do indeed learn the axioms first and only then the consequences. So a leap of faith is indeed required there. But there is not really a better way of teaching mathematics. As long as the axioms are motivated, I don't see a problem.

The essential thing is when doing research yourself. In research, there is no such thing as blindly accepting axioms. In fact, you first solve the problem and only then look at how you would present it. So in research, choosing the axioms and definitions come last.

There are very very few mathematicians who choose axioms for fun and then see what they can deduce. That is just not workable. I'm sure it happens, but those mathematicians won't really amount to anything.

all those non-motivated definitions

That is not an argument against axiomatics and deductive reasoning. It's just that the particular presentation of the mathematics is bad. It is perfectly possible to start with axioms and definitions and to have everything motivated clearly. For example, Carothers' real analysis is one of those books for real analysis. Artin's algebra is one of those books for algebra.

 

[jostpuur]

These are supposed to be different things:

http://en.wikipedia.org/wiki/Inductive_reasoning

http://en.wikipedia.org/wiki/Mathematical_induction

Show me one theorem proven with inductive reasoning, and you win.

No, I wouldn't be winning anything.

Are you guys sure you can afford to get this philosophical? Conserning "showing" things, I would like to see if somebody can show me a person who has learned Galois theory by reading important definitions and theorems of the Galois theory. That would be something.

 

[microsansfil]

Deductive reasoning is incapable of yielding essentially new knowledge,plausible reasoning

Polya 's plausible reasoning is also a deductive reasoning issued in an uncertain environment. The conclusions can be obtained by the use of probabilities, and more specifically by using Bayesian inference.


All probability calculation (including Bayesian) incorporates a process of deduction, because any calculation is is a process of deduction.

As means, the deduction (calculations) are everywhere. As methods of reasoning, the difference is not in the calculation, but in the interpretation of the calculations.

Deduction as a method relies on a notion of consistency. We test whether a set of proposals is consistent or not, and we reject (deduce, meaning remove) any game incoherent. This is the method of Sherlock Holmes : "when you have eliminated the impossible, whatever remains, however improbable, must be the truth". impossible understood as "inconsistent with the known", "known" is set as certain.


The induction as a method intended to evaluate likelihood without accepting impossibility.

Patrick

 

[whyevengothere]

Sure, that is completely true. But deductive reasoning is absolutely essential in checking the results and presenting them. I have plausible ideas all the time, and only 1% of them ever works if I check it rigorously. So axiomatics and deductive reasoning are extremely important in mathematics and are responsible for making math correct and trustworthy.

In teaching of math, both deductive and plausible reasoning should get attention, and both do get attention. Students should absolutely learn about logic and deductive reasoning because they should know how to check that their arguments are correct. It is only a deductive argument that is accepted in math (rightfully so), so the students should learn it. That said, plausible reasoning also occupies a large part of learning mathematics. In fact, whenever you do exercises, the teacher will teach you about plausible reasoning (if he's any good).
Not at all. Axioms can and should be motivated. Blindly accepted axioms with the hope that something fruitful comes out is the wrong way of doing thing.

That said, when learning math, we do indeed learn the axioms first and only then the consequences. So a leap of faith is indeed required there. But there is not really a better way of teaching mathematics. As long as the axioms are motivated, I don't see a problem.

The essential thing is when doing research yourself. In research, there is no such thing as blindly accepting axioms. In fact, you first solve the problem and only then look at how you would present it. So in research, choosing the axioms and definitions come last.

There are very very few mathematicians who choose axioms for fun and then see what they can deduce. That is just not workable. I'm sure it happens, but those mathematicians won't really amount to anything.
That is not an argument against axiomatics and deductive reasoning. It's just that the particular presentation of the mathematics is bad. It is perfectly possible to start with axioms and definitions and to have everything motivated clearly. For example, Carothers' real analysis is one of those books for real analysis. Artin's algebra is one of those books for algebra.

What books take the opposite approach? Rudin's book on analysis?the Bourbaki books?
By the way what do you think about V.A Zorich's books on analysis ? they make it very easy to work with physical problems ,and the exercices are very hard. I've read the 3 first chapters of Artin's book,it is indeed very good.

 

[micromass]

What books take the opposite approach? Rudin's book on analysis ,the Bourbaki books?

Yes, and in my opinion those books are horrible. But the problem isn't with axiomatics, but with the writing that is not motivating the theory. I don't mind hard books, but Rudin is hard for the wrong reasons. Sadly enough, most analysis classes do use Rudin for some weird reason.

By the way what do you think about V.A Zorich's books on analysis ? they make it very easy to work with physical problems ,and the exercices are very hard.

The books are very good indeed, they're also one of my favorites.

Choosing books is very personal. I think it is very possible to find books you will like, but you'll usually have to search very hard.

 

[DivergentSpectrum]

In my opinion its a fluid approach. I find it extremely annoying when I go on wikipedia and they throw the abstract group theory at me when I am looking for some simple math concept, lol. I was also horrified when I was watching youtube lectures on electromagnetism(MIT provided, believe it or not) and the professor"proved" gauss law for the flux of the electric field by generalizing the case where a point charge is in the center of a sphere, without mentioning a thing about the apparent inconsistency when you apply the divergence theorem and how the dirac delta function is used to deal with this problem.

In all honesty, the second pissed me off more.