Proof mathematics subjects at University

[pivoxa15]

I have done all the first year mathematics subjects at University and now doing some higher level ones such as a couse called Number Theory - although this course is only an introduction to the subject, I still find it tough mainly from following the proofs in the course. In the past, I have got through math pretty much by computations but now I realize that proofs are much more significant in mathematics so I really want to master it.

The trouble is, I find it difficult. Anyone have any advice to how I can become more comfortable with them? Is it simply the case of practice or are there certain methods of learning?



Thanks

 

[quasar987]

If I were you, I wouldn't worry if you can't follow the proofs of your teachers. The thing with doctors and older teachers in general*, is that they forgot where they had difficulties when they were learning the subject, ans thus don't emphasize the important points. For them, their proofs are self evident and they have trouble imagining that they are not cristal clear to everyone. They tend to skip explaining the logic btw each step of their proof, which makes each of them kind of little puzzles to solve for the students. Of course, this solving cannot be done LIVE in the course, so after a while, most students are loss and just copy what's on the blackboard mindlessly.

Of course you have better chances of sucess trying to follow in class if you've read on the subject before class, but this is almost always impossible to do given the overloaded schedules we have.

*This is of course only how I perceive things based on my own life experiences.

 

[matt grime]

If I were you, I wouldn't worry if you can't follow the proofs of your teachers. The thing with doctors and older teachers in general*, is that they forgot where they had difficulties when they were learning the subject,

i doubt any of them had any difficulty in mastering proofs in basic courses, which may be more of a problem.

in any case, you aren't supposed to understand all proofs when you first read them and if you think you ought to be able to then you will only ever have headaches. you need to go away, read, reread, write out the notes again (if not rewrite the rewrite) and attempt to understand the logic yourself.

It is primarily a case of practising and noting that there aren't that many proofs available to you. For instance, if you are doing a problem in number theory and it starts: suppose x and y are coprime integers... then it is almost certain that if you start by noting that this implies there are integers u and v such that ux+vy=1 you will probably find the answer quite quickly. (basic analysis proofs are even easier though few undergraduates ever believe me).

it is also important to note the difference between something being an 'if' statement and an 'if and only if' statement (aka 'iff').

 

[mathwonk]

My guess is that many students have never studied basic mathematical logic.

Proofs use these fundamental principles over and over.

e.g. to prove that a statement like: " every continuous function is differentiable" is false, it is adequate to find a single continuous function which is not differentiable.

on the other hand, to prove that every differentiable function is continuous, one must show how the property of differentiability forces the property of continuity.

to show that every continuous function on a closed finite interval is bounded, it is enough to show that an unbounded function on a closed finite interval cannot be continuous.

In my own personal case, I have not forgotten the difficulties involved here, but was lucky enough to have studied logic in high school from a basic book on foundations of math then used in colleges, "Principles of Mathematics" by Allendoerfer and Oakley.

This, "leg up" enabled me to start college calculus on the right foot, but it was still very hard.

I do tend to forget that many students have never learned logic, as it seems so intuitive. E.g. the student who says that "I'll jump off the 10 meter board if you will", thinks he is telling the truth even if neither jumps off.

But that same student claims not to understand how a mathematical statement whose hypothesis is false can be true. Why is this so common?

 

[honestrosewater]

Maybe proving some things you already know are true would help you become more comfortable with the process. For instance, you're already familiar with the real field, so you could try to prove some basic things about it. You can find the field axioms several places:
http://mathforum.org/library/drmath/view/51925.html
http://mathworld.wolfram.com/FieldAxioms.html

Using those, try to prove the statements in sections 4 and 5:
http://www.math.uiuc.edu/~rezk/number-systems.pdf

Maybe someone else has better resources. These are just the first ones I found in a quick search.

I just happened upon a http://www.math.uchicago.edu/~eugenia/proofguide/proofguide.pdf .

 

[mathwonk]

for instance to prove every function cont on a finite closed interval is bounded, one might begin by saying: " suppose it is not bounded.." This already puzzles some students.

The basic rule that "A implies B" is true if anfd only if the statement "notB implies notA" is true, is not familiar to them. It was the old proof by contradiction that used to be taught for thousands of years in euclidean geometry in high school, where most profs studied (some of us thousands of years ago).

Diatribe begins roughly now:

But today's brilliant pedagogues in high school have replaced this kind of reasoning with vacuous courses on computation of areas of circles, or AP courses in which people try to memorize all the answers to some standard test, rather than learning how to reason.

Older college profs are perhaps to some extent still slightly unaware of how empty is the current high school curriculum in the US.

Even AP courses for example focus on memorizing factual content rather than understanding principles for solving problems.

On a typical AP AB calc test a few years ago, there was not a single proof question.

So as college professors have spent 20-30 years teaching, the high schools have spent that time watering down the high school math curriculum, so that entering students have nothing like what used to be standard knowledge.


The fundamental prerequisite for doing a proof is to be able to understand:

1) what does the statement to be proved actually say?

2) what would it mean for that statement to be true, or false?


how can a student grasp these things if he/she cannot even write a sentence with a verb in it? this is a real challenge for some of our charges. our stats show that scores in our basic math courses are correlated not with math SAT scores, but with scores on verbal SAT. So which SAT has been targeted for refurbishing? The verbal SAT, from which the most useful section, namely analogies, is to be removed I understand.

It is very difficult for us to keep up with the demands of teaching new students when the traditional background is watered down in high school almost every year and replaced by worthless lists of topics to be covered, such as on AP tests.


Teachers, please don't teach high school (or other) students to memorize particular dates, formulas, or other facts, show them how to understand the statement of a problem, and then how to reason from that understanding towards a solution.

Finally teach them to write up that solution in complete sentences.

Then heaven will come on earth, and all our students will be able to do proofs.

And perhaps they will then observe that Hans Blick was right after all. :zzz:

recommended reading: Harold Jacobs' Geometry, a brilliant, amusing, well illustrated, well written, deeply substantive, high school text.

 

[honestrosewater]

Maybe more than anything else in logic, learning - and practicing with - a natural deduction system could help in writing proofs and discovering some common proof techniques. These systems are designed to follow people's natural reasoning processes, and they're very easy to learn. It's just basic 'what can you infer from what' stuff. I don't think that mindlessly applying rules can get you very far with these systems, rather you actually need to understand what you're doing. But maybe others here have had bad experiences with this.?

I like the system in Copi & Cohen's Introduction to Logic, which should be easy to get ahold of (check your library). There are also several places on the internet that present a natural deduction system:
http://tellerprimer.ucdavis.edu/
http://www.mathpath.org/proof/proof.inference.htm
http://www.danielclemente.com/logica/dn.en.html
http://en.wikipedia.org/wiki/Propositional_calculus#Inference_rules

These systems work like so:
You're given some inference rules and possibly some replacement rules. For example, some Inference Rules:
Modus Ponens: Given P implies Q and P, you may infer Q.
Reductio ad absurdum: If you can derive Q and not Q while assuming P, you may infer not P.
Replacement rules:
You can replace any instance of P implies Q with (not P) or Q and vice versa.
For each proof, you're given a conclusion and a (possibly empty) set of premises, and you must choose which rules to apply in what order in order to derive the conclusion. Obviously, it can end up taking a very long time to derive your conclusion if you just randomly choose and apply rules.

 

[matt grime]

my oft stated displeasure in over teaching logic (not that i disapprove of logic at all, or foundational maths) comes from the fact that the only necessary skill is to be sensible, and that learning when a convoluted statement is a tautology has no bearing on ever proving anything except in formal logic. we shouldn't need to teach at university level that to show A implies B is false it suffices to find a case when A is true and B is false. That should be patently obvious to anyone doing a degree in maths. Besides which, that doesn't ever actually help a student find such a case which is what is important.

 

[honestrosewater]

Perhaps just going through the list of rules and understanding why eash is valid would be helpful for someone who hasn't bothered to give them much thought? It would at least make their 'common sense' reasoning explicit.

 

[matt grime]

would it? i obtained all my maths qualifications up to masters level and was well into my phd before it came to my attention that A implies B is the same as not(A)\/B. can't say it bothered me that i'd never thought of that until required to teach it.

 

[ComputerGeek]

(basic analysis proofs are even easier though few undergraduates ever believe me).

I believe you :-)

my basic analysis class was so much easier (in hindsight) than some of the proof I did in Abstract Algebra and I will be taking basic Number Theory this fall. I am waiting with ambivalence at this time.

BTW, how long does it take to get the amount of experience to recite information as you do? I have not had calculus in almost 2 years and I have gotten very rusty with it and fear encountering some of the ideas again in grad school (I am sure it will come back to me, but still)

 

[ComputerGeek]

Teachers, please don't teach high school (or other) students to memorize particular dates, formulas, or other facts, show them how to understand the statement of a problem, and then how to reason from that understanding towards a solution.

Finally teach them to write up that solution in complete sentences.

I plan to teach problem solving, come hell or high water. what is interesting is that there is a strong movement in public schools now that focus on getting students to think critically and get better at comprehending information. that is something that math is uniquely qualified to introduce to students due to the required discipline of thought and reasoning that math requires.

 

[matt grime]

BTW, how long does it take to get the amount of experience to recite information as you do? I have not had calculus in almost 2 years and I have gotten very rusty with it and fear encountering some of the ideas again in grad school (I am sure it will come back to me, but still)

i can probably only recite elementary analysis proofs because i teach them every year; i have no interest in analysis anymore apart from that. however, years of experience makes you realize how few methods of proof you need. even the great hilbert is alleged by one author whose name escapes me to have reused the same 6 ideas to prove his most astonishing ideas.

however, unlike the rest of these forums, the questions asked here are quite elementary to a mathematician, and i am not the first to mention this difference between the maths forums and the physics, chem, or biology forums; this i think reflects two things - the US education system wherein lots of maths is taught as a background for all science subjects and it is students in those classes who post here predominantly, and the fact that advanced mathematics is a mystery even to those who are practitioners in its dark arts. you get few questions here of the level of a US graduate course.

 

[honestrosewater]

would it? i obtained all my maths qualifications up to masters level and was well into my phd before it came to my attention that A implies B is the same as not(A)\/B. can't say it bothered me that i'd never thought of that until required to teach it.

I don't know - I'm not very far ahead of the OP. That's why I was careful to say maybe and ask for others' opinions. But for your example - how then do you reason that (P -> Q) <=> (~Q -> ~P)? Isn't that a helpful thing to know? I think it's easy to realize that they are both equivalent to (~P v Q). Also, if you're trying to prove a statement of the form (~P v Q), and you're already familiar wth proving (P -> Q) via a conditional proof, you would know right away how to approach your proof. To someone who's just starting out, this kind of stuff might not be so obvious. A conditional proof may not be the first thing that comes to mind when someone sees (~P v Q).

 

[ComputerGeek]

i can probably only recite elementary analysis proofs because i teach them every year; i have no interest in analysis anymore apart from that. however, years of experience makes you realize how few methods of proof you need. even the great hilbert is alleged by one author whose name escapes me to have reused the same 6 ideas to prove his most astonishing ideas.

however, unlike the rest of these forums, the questions asked here are quite elementary to a mathematician, and i am not the first to mention this difference between the maths forums and the physics, chem, or biology forums; this i think reflects two things - the US education system wherein lots of maths is taught as a background for all science subjects and it is students in those classes who post here predominantly, and the fact that advanced mathematics is a mystery even to those who are practitioners in its dark arts. you get few questions here of the level of a US graduate course.

so, basically the more you teach it the more you know it. well, I do not feel so inadequate then at this point in my life.

 

[pivoxa15]

Thanks for your suggestions.

A thing I noticed was notations. Maybe most of my trouble comes from not fully understanding what the notations mean which hinder my understanding of the proof. And one small misunderstanding or lack of understanding here and there in a proof could be disastrous, leaving me in total confusion and frustration.

 

[honestrosewater]

A thing I noticed was notations. Maybe most of my trouble comes from not fully understanding what the notations mean which hinder my understanding of the proof.

Which notations?

 

[matt grime]

I don't know - I'm not very far ahead of the OP. That's why I was careful to say maybe and ask for others' opinions. But for your example - how then do you reason that (P -> Q) <=> (~Q -> ~P)?

Easily: if I know that P implies Q, and I know not(Q), then not(P) must follow, otherwise we must have P, but that would imply Q, contradiction since we have not(Q). Hence P implies Q means that not(Q) implies not(P), the reverse implication now follows by symmetry. This is just the reason why we declare ordinary deductive logic to have the rules it does.

 

[honestrosewater]

Easily: if I know that P implies Q, and I know not(Q), then not(P) must follow, otherwise we must have P, but that would imply Q, contradiction since we have not(Q). Hence P implies Q means that not(Q) implies not(P), the reverse implication now follows by symmetry. This is just the reason why we declare ordinary deductive logic to have the rules it does.

Where did you come up with the rules that you used? Did you ever doubt that your 'natural' reasoning was correct? I mean, I'm not doubting you; I'm curious. How can someone do math without having the rules clearly set out ahead of time?

 

[arildno]

Where did you come up with the rules that you used? Did you ever doubt that your 'natural' reasoning was correct? I mean, I'm not doubting you; I'm curious. How can someone do math without having the rules clearly set out ahead of time?

Actually, the rules for which inferences is to be regarded as logically valid should be part of the explicit set of axioms within the particular maths you're doing. It's just that most won't bother to list them; I guess..

 

[honestrosewater]

Actually, the rules for which inferences is to be regarded as logically valid should be part of the explicit set of axioms within the particular maths you're doing. It's just that most won't bother to list them; I guess..

Yes, good point. So am I the only one who finds that it's easier - and puts you on firmer ground - to just state the rules that you're allowed to use?

 

[arildno]

Yes, good point. So am I the only one who finds that it's easier - and puts you on firmer ground - to just state the rules that you're allowed to use?

A good point as well!

However, I think a convention has established itself that logicians state&study sets of rules of inference, whereas mathematicians just use the "natural" system, unless otherwise specified.

I might be wrong on this, of course; the math wizzes will know better.

 

[honestrosewater]

Well, this isn't directed at matt or anyone else in particular. I just don't understand the idea that people should be left to figure out the rules on their own. If the rules are obvious, it won't hurt to state them. If the rules aren't obvious, they should be stated. ??

 

[matt grime]

Where did you come up with the rules that you used? Did you ever doubt that your 'natural' reasoning was correct? I mean, I'm not doubting you; I'm curious. How can someone do math without having the rules clearly set out ahead of time?

i understand your point, but the rules are the obvious and the minimal ones such as not(not(A)) is A, and that's about it, indeed that is all I used along with the notion of what deduction means. About the only other one we need is A implies B and B implies means that A implies C the associativity of deduction but again that is simple elementary logic we learn possibly even before school.

there are many subtle points in logic, almost none of them is of interest in other parts of 'functional mathematics' by which i mean the notion of practising mathematics to any reasonable level. Another one that we can and should overlook is what a set actually is.


as another example, until i was forced to teach it against my will i had never thought that false implies true is true; i had also never thought it false or for given it one second's consideration at all. It has no bearing on mathematics. If I want to show that f continuous on a compact subset of R implies f is uniformly continuous I would try to show how we can deduce the antecedent from the hypothesis or whatever the language is. I would never consider the case where the antecedent was false.


in short what A implies B is when A is false is not of direct interest in proving that A implies B. It may be if it we are wondering about the converse.

 

[roger]

you mean when a implies b and b implies c ?

 

[roger]

As far as axioms go, how do you know when to stop ?

That is to say, at the most basic fundamental level, we make certain rules but how do we know when to stop proving ?

 

[mathwonk]

well i agree the rules should be stated, but they are stated in traditional geometry courses, but as i said, many high schools stopped teaching that subject in favor of AP calc, a stupid move in my opinion.

that said, the rules are also obvious:

e.g.

if you say "I'll jump off the high board if you do"

Isn't that the same as saying " if I do not jump off the high board, then you didn't either".



or if you say every differentiable function is also continuous"

isn't that the same as saying " any function that is not continuous is also not diffentiable".

or "if I leave the room, then I cannot lock the door from the inside",

is the same as "if I locked the door from the inside, then I did not leave the room".

or "if you make all A's I'll give you a transam" ism the same as "if i do not give you a transam, then you did not make all A's".


These are simple examples of "A implies B" if and only if "notB implies notA",

and they should be relatively obvious upon reflection.

most school sdo teach these rules of logic now, to make up for the poor high school training now provided, but unfortunately they do so in later years, like junior year, instead of freeshman year.

the reason is we have people in engineering schools wanting their students still to ahve in first eyar and not to take up a whole semester rpoviding makeup for missing high school courses,

it is very hard to constantly keep taking up in college the slack provided by our high school system's constant decline, but we are trying.

whats wrong with the student taking some initiaitve and reading a book on the topic, i did that as a high school senior.

we have recommended several already.

 

[matt grime]

that is one of the more important aspects in which logic and foundational mathematics are interesting. no axiom is true in and of itself and we are free to pick whatever we choose as our axioms. if we add the axiom of choice then every vector space has a well defined notion of basis, but then banach tarski is true.


we use whatever we feel necessary, to be honest. try reading gowers' a very short introduction to mathematics.

 

[roger]

so you mean the axioms are relatively true ?

 

[mathwonk]

math is about proving things like if A holds then B does too.

we do not care whetehr A is true or not, that is up to the suer.

i.e. a statements like that is useful to anyone ni whose world A actually is true.

we prove statements involving many axiom sysstems so lots of people in different worlds can use them.

you bring the list of axioms you expect to hold in your world and we will tell you what you can deduce from them for your own use.

like in europe power runs at something 220 volts so people make devices that work that way, in the US power is at 110 volts so we make also devices for them.

different worlds satisfy different axioms systems.

 

[chronon]

Yes, good point. So am I the only one who finds that it's easier - and puts you on firmer ground - to just state the rules that you're allowed to use?

But eventually you have to appeal to something being obvious: http://www.lewiscarroll.org/achilles.html

 

[Doodle Bob]

Yes, good point. So am I the only one who finds that it's easier - and puts you on firmer ground - to just state the rules that you're allowed to use?

Actually, that's why they teach (or are supposed to teach) logic and truth tables in high school (and later on in college). I don't know about most other states, but these topics are still in Ohio's mathematics standards for K-12 education.

 

[mathwonk]

I am surprized that truth tables are on the high schools standards, and impressed, at elast if they get taught.

are they not only in the standards but also actually taught there? in my state, such lists of standards are sometimes very overly optimistic lists of everything some one thinks will look good.

but in my state the teachers are not at all prepared to teach most of what is on the standards. in fact even the booklet designed to help them prepare for the teachers' test is written by people who also do not understand most of the topics.

(I know, I reviewed it as a professional consultant.)

of course I am sadly aware my southern state is a weak example. I am scheduled to teach truth tables to college juniors this fall, and in the past such students have never seen them before.

 

[honestrosewater]

there are many subtle points in logic, almost none of them is of interest in other parts of 'functional mathematics' by which i mean the notion of practising mathematics to any reasonable level. Another one that we can and should overlook is what a set actually is.

Which is the reason for setting out the rules: there's no arguing over what a set 'actually' is - a set is whatever the definitions or rules say it is. IMO, what you mean is a matter of philosophy, not logic.

as another example, until i was forced to teach it against my will i had never thought that false implies true is true; i had also never thought it false or for given it one second's consideration at all. It has no bearing on mathematics. If I want to show that f continuous on a compact subset of R implies f is uniformly continuous I would try to show how we can deduce the antecedent from the hypothesis or whatever the language is. I would never consider the case where the antecedent was false.


in short what A implies B is when A is false is not of direct interest in proving that A implies B. It may be if it we are wondering about the converse.

I guess that's just what happens when you formalize the rules that you're already using. I gather you don't think that formal logic is unnecessary, but maybe you think it is a necessary evil?
Those .\overline{9} = 1 threads may serve as an example. You first try to explain things through common sense and mathematical reasoning and resort to the formalism and foundations when nothing else works?

 

[honestrosewater]

But eventually you have to appeal to something being obvious: http://www.lewiscarroll.org/achilles.html

Actually, that's a good example of why you need to just set out the rules in the first place. Then if you want to question those rules, you may do so in another system.

 

[matt grime]

I gather you don't think that formal logic is unnecessary, but maybe you think it is a necessary evil?

evil? not at all; i find a logic a very useful and interesting subject and have been even known to read things on O-minimal structures and listen to talks by logicians.

I just think it unnecessary to teach logic to people upon entering university. the basics, all that is needed, should be naturally obvious and even if not they will become apparent quickly if we teach them properly what a proof is.

but then i may be expecting too much of people when i think that anyone doing a degree in mathematics ought not to be a moron: why on Earth should I have to prove to someone that A implies B and B implies C implies A implies C? If you do A, B happens, if B happens then C happens, thus if you do A C happens, come on, it's not rocket science. i mean when ever in mathematics do you ever have to prove anything by appealing to some ridiculous complicated statement being a tautology? seriously, beyond not not A is A, and A implies B being obviouly not(B) implies not(A), as well as the associativity of implication what on Earth do we need from logic in order to start teaching matheamtics? perhaps a simple idea of negation of statements and a little quantifier stuff, but again it should be obvious that in order to show that for all X, P(X) implies Q(X) is false by finding a counter example, or more importantly, it should be obvious after one example in their course that this is how it can be done.

there is nothing at this level of utility that needs to be treated as a speciality of logic, it is just elementary maths that you pick up quickly. drawing attention to it is not necessary. i still get students who don't understand that writing out a truth table for all possibilitise of A and B doesn't prove that A implies B in any particular case.

The difficult and interesting stuff is not teachable to people who are struggling to comprehend this weird thing that is the first year of degree level mathematics.

I would no more give an deep introduction to groups rings and modules to a first year as a prelude to explaining what euclid's algorithm than I would enter a deep discussion on predicate calculus with someone before explaining what a sequence is and what it means for it to converge (this is not a vacuous example and is exactly what I am required to teach to my first year tutees).

 

[roger]

Regarding the associativity of implication, matt referred to something which has nothing to do with maths, by saying if you do a , b happens etc..and implied it was easy to understand.

Its actually no more obvious than why 1+1=2 in my opinion

 

[Eratosthenes]

Anyone have any advice to how I can become more comfortable with them? Is it simply the case of practice or are there certain methods of learning?

You should be comfortable proving things different ways. Here are three different ways to prove the same thing that you will find extremely useful I think. I'll try to explain them as best as I can. A good book would be better of course!

Prove that if x is an even integer then x + 9 is odd.
For the examples below, let A and B denote the following.
A: x is an even integer.
B: x + 9 is odd.

Method 1(sometimes called direct proof)
If A holds then B must follow. So we are showing that if x is an even integer, then x + 9 must be odd.

Proof
Suppose x is even. Then x = 2k for some integer k. Then x + 9 = 2k + 9 = 2k + 8 + 1 = 2(k + 4) + 1. Since k + 4 is an integer(you can state reasons if you want here, closure etc, doesn't matter I think), then x + 9 is odd.


Method 2(sometimes called proof by contradiction or indirect proof)
We assume that it is not true that if A holds, then B must follow. Then we reach a contradiction to show that our assumption was false, which means the negation of our assumption is true.

Proof
Suppose x is even and x + 9 is even(this is the negation of "if A, then B").
Then x + 9 = 2k for some integer k. Then x = 2k - 9 = 2k - 10 + 1 = 2(k - 5) + 1, where k - 5 is an integer. Thus x is odd. Now we have a contradiction. So the negation of our assumption must be true, thus x + 9 is odd.


Method 3(sometimes called proof by contraposition)
Here we use the contrapositive of "If A, then B." Which is: If not B holds, then not A must follow. So we are trying to show that if x + 9 is not odd, then x is not even. This is the same as trying to show that if x + 9 is even, then x is odd.

Proof
Suppose x + 9 is not odd, hence even. Then x + 9 = 2k for some integer k. Then x = 2k - 9 = 2k - 10 + 1 = 2(k - 5) + 1, where k - 5 is an integer. Thus x is odd.

If you ever run into something like prove A if and only if B, where A and B are statements you need to do two things.
1. You must show that if A holds, then B must follow.
2. You must show that if B holds, then A must follow.


The most useful example is method 1. However some proofs are easier with other methods, so it's good to be familiar with all the them. Wikipedia should have better explanations of each method and your number theory book should provide lots of practice:)

Induction is another method you should be really familiar with if you are taking Number Theory. A good exercise in proof by contradiction is to prove the Principle of Mathematical Induction. Anyways I hope there are no really bad mistakes in those examples, the last thing I want to do is confuse you more. Goodluck to you!

 

[matt grime]

Let me see, so if it is true that when i press this here button a bell rings, and if it is true that when the bell rings my pavlovian dog drools, then why is it not obvious that pressing the button ends up with me having a saliva dripping pavlovian dog?

 

[roger]

matt, wouldn't you say this is heading towards philosophy rather than having anything to do with maths ?

I recall, you yourself explicitly advised me to disregard and steer clear of physical examples or referring to the 'real' world .

And I can't say the example above is obvious to me actually.

 

[matt grime]

It has nothing to do with philosophy, and it is perefctly valid to use real world examples since we are fixing what rules of deduction we will use in maths which are based upon our experiences in the real world. In fact we are discussing exactly what it is necessary to specify in our system of logic. the choices for that will be based upon what is 'obviously true'. just as it is perfectly reasonable to look at some real phenomenon to figure out what it is we wish to model, say.

We are not considering maths of the kind my advice referred to and in the sense in which the advice was given and you ought to realize that.

 

[roger]

Well if it is to be based upon what is 'obviously true', then that again is something debateable.

And in any case, what do you define as the real world ?

this is why I said it could be heading towards philosophy.

And when we do look at phenomenon, and try to model it, we run into trouble anyway, so it goes to show, looks can be deceptive.

 

[honestrosewater]

matt,
I think I finally understand you, and you've changed my mind about what is appropriate at which level. Since people often post questions about adapting to proofs, I think it would be nice to have a list and brief explanation of the useful rules for them. Do you want to add any more rules to your list?

roger,
I don't see what you're getting at. In matt's example, the subject matter of the atomic propositions has no bearing on the argument's logical validity, which is what he was highlighting.

Let me see, so if it is true that when i press this here button a bell rings, and if it is true that when the bell rings my pavlovian dog drools, then why is it not obvious that pressing the button ends up with me having a saliva dripping pavlovian dog?

P = I press the button
Q = The bell rings
R = Pavlov's dog drools

If P, then Q. If Q, then R. Thus, if P, then R.

Does it make sense to you that if x = y and y = z, then x = z? How about if x > y and y > z, then x > z?

 

[matt grime]

Of course I should have said transitivity when i was trying to illustrate my examples.

Also, roger, when I say something in this argument is "obviously true" I mean "I obviously want it to be true because that is nice" or soemthing. I don't mean it has an easy proof, indeed it has no proof really since it is one of those axioms we are going to have in our system.

 

[honestrosewater]

Also, roger, when I say something in this argument is "obviously true" I mean "I obviously want it to be true because that is nice" or soemthing. I don't mean it has an easy proof, indeed it has no proof really since it is one of those axioms we are going to have in our system.

Well, not to confuse things - you're talking about math and I about logic, but here are the proofs in two natural deduction systems, with minimal explanation. I'm quite curious to see if they make sense to anyone.

In the system that I'm used to, the argument to be proven is actually one of the rules, so the proof is quite simple (anyone can look up the names I use here - Edit: there are explanations of them further down the page):

1) P -> Q [premise]
2) Q -> R [premise]
3) P -> R [1, 2, Hypothetical Syllogism/QED]

This system has no conditional proof (you can't introduce assumptions out of nowhere) and I can't see a way to prove the exact argument without the HS rule, so I'm adding another premise: P.
1) P -> Q [premise]
2) Q -> R [premise]
3) P [premise]
4) Q [1, 3, Modus Ponens]
5) R [2, 4, Modus Ponens/QED]
This just shows that (P -> Q) and (Q -> R) connect in such a way that if P is true, then R must also be true.

The other system, which I'm not familiar with, does have a conditional proof rule, so I can prove that [((P -> Q) & (Q -> R)) -> (P -> R)] is a tautology by deriving it from nothing, i.e., the empty set of premises (and hopefully not make any technical blunders).
1)) (P -> Q) & (Q -> R) [hypothesis - I assume this in order to derive the tautology, note that it is nested]
2)) P -> Q [1, Conjunction Elimination]
3)) Q -> R [1, Conjunction Elimination]
4))) P [hypothesis - I assume this in order to derive (P -> R), nested again]
5))) Q [2, 4, Modus Ponens]
6))) R [3, 5, Modus Ponens]
7)) P -> R [4, 6, Conditional Proof]
8) ((P -> Q) & (Q -> R)) -> (P -> R) [1, 8, Conditional Proof/QED]

(I edited some unnecessary steps I used the first time.) This says the same thing: Given (P -> Q) and (Q -> R), if we assume that P is true, (P -> Q) gives us Q. Now that we have Q, (Q -> R) gives us R. So (P -> Q) and (Q -> R) give us (P -> R). Pretty straightforward, I think. The formalism may turn some people off, but it was comforting to me to see everything laid bare. Of course, I'm quite a strange cookie.

 

[roger]

honestrosewater, if the subject matter has no bearing on the logical validity, then why did matt pick that particular example ?

And what relevance does if x=y , y=z then x=z have to this ?

 

[matt grime]

I picked that example for no particular reason. Any example would have done:

If i have a bad fall down the stairs then i break my leg. If I break my leg I go to hospital, if i go to hsopital with a broken leg they;ll put it in plaster, hence if I hava a bad fall down stairs I will end up with my leg in plaster.



If you can't see what honestrosewaters transitivity of equality has to do with it, or any transitivity tehn can i ask you to go away and think abuot this carefully?

another examplei s that i can travel by train from X to Y and I can travel by train from Y to Z then I can travel by train from X to Y.

 

[roger]

but surely isn't there a difference between the transitivity of equality and that of logic ?

they are two separate issues ?

(that should be from x to z not x to y.)

I have referred to your website, and read your thoughts on these matters, you have asserted that maths has nothing to do with the real world, so I simply can't understand why now you chose to appeal to a real world example to demonstrate a mathematical concept ?

 

[matt grime]

i am not demonstrating a mathematical concept. your arguments seem quite specious and merely designed to provoke me into response. objects in mathematics are idealized, that doesn't mean that we cannot reason with them as if they were part of the real world, for heaven's sake, any more than results in combinatorial graph theory cannot be applied to assigning lecture times.

in any case you seem to be entirely mising the point of this argument which is that common sense is all that is requierd to understand what deductive reasoning is required to start doing mathematics. ignore the maths for one second, do you understand simple deductive reasoning in the eral world? If i buy that car I will leave myself with less than 1,000GBP in the bank, if I have less the 1,000 GBP in the bank then I can't afford that cruise in the caribbean, hence if i buy the car i can't go on the cruise. i am making no assertions about mathematics at all, am I? or claiming to? now, why when we start thinking of mathematics must i use a different set of deductive rules? (we may wish to as the constructivists might argue) but that isn't what we're talking about. we are wondering how it is that people entering university are unable to make simple logical deductions be they about mathematics or anything else it would appear if your responses are anything to go by.

 

[meckano]

I didn't read all posts, please excuse repetition.
I find you have to care.
I love math.s and did well in school, but could have done better if I cared about it.

Try this:
Create a situation where you want THE answer on the first try.
- If you don't get THE answer, imagine catastrophe; Public embarrasment, Mechanical failure/costs... whatever.
So, you need to prove to yourself that it is the right answer. You're in the woods, no internet, no towns, nobody.
I had found that it is What I am proofing with that I did not understand completely.
I had to go over what I did not understand and IT's proofs.
Don't stop till you get, but you won't stop cause you care.

I consider the above to be the definition of: learn by rote, aka: keep redoing it till you get it.
- Just keep looking at if from different angles.
Like in my post in 4 dimensions, I like 1, so I checked to see how a circle with a diameter of 1 is related to other circles. (I was lucky to have heard about the power of 1 in math, somewhere, some time ago.)
- Grab on to a reason to learn, a mathematical reason for maths. So wanting to teach it won't be a helper in understanding math.s, although it will be a helper to keep you going on the days you walk around scratching your head wondering if you'll ever Get-it.

:) hope this helps.