Not sure how to deal with the abstract nature of maths

[Logical Dog]

I have been reading some books (Allendoerfer, principles of math, Zakon, series of mathematical analysis, R. Courant, what is mathematics).

I have learned that some of the basic fundamental, the msot bare bones of mathermatical concepts and definitions have to remain undefined. These are set, element, point, line, proposition, true, false.

Other technical terms are later defined in terms of these.

What do I make of this? we have an intuitive guide to all these terms and certainly treat them as we know what they are. We were discussing this with friends, and I simply stated one of courants views roughly as being the student should not be concerned with the philosophy or nature of mathematical objects, rather it is the relations and operations between them that are more important.

It still blows my mind somewhat.

 

[Lucas SV]

You will find the axiomatic method again and again as you learn more maths, although usually you will have the defined terms already, in terms of set theory. For example, the definition of topology relies on sets. It is nevertheless axiomatic in the sense that it gives axioms which distinguish topologies form non-topologies. Similarly Zermelo Fraenkel set theory gives axioms which distinguish sets from non-sets.

However, to really understand a subject, you need to have built something out of the axioms. Then you will appreciate the axioms much more, compared to when you have seen them for the first time.

 

[Logical Dog]

You will find the axiomatic method again and again as you learn more maths, although usually you will have the defined terms already, in terms of set theory. For example, the definition of topology relies on sets. It is nevertheless axiomatic in the sense that it gives axioms which distinguish topologies form non-topologies. Similarly Zermelo Fraenkel set theory gives axioms which distinguish sets from non-sets.

However, to really understand a subject, you need to have built something out of the axioms. Then you will appreciate the axioms much more, compared to when you have seen them for the first time.

According to my limited knowledge, an axiom is an initial proposition we assume to be true, despite having no such knowledge, and we can choose freely our axioms. It is just very abstract at some level for me, I am not able to explain the feeling well.

 

[Lucas SV]

According to my limited knowledge, an axiom is an initial proposition we assume to be true, despite having no such knowledge, and we can choose freely our axioms. It is just very abstract at some level for me, I am not able to explain the feeling well.

That is right. Then you can take a set of propositions and call it an axiomatic system. In set theory and logic, you may define what it means for an axiomatic system to be consistent. You may also define what a proof means. I haven't actually red this, but it may help: https://archive.org/details/IntroductionToTheFoundationsOfMathematics.

 

[Logical Dog]

That is right. Then you can take a set of propositions and call it an axiomatic system. In set theory and logic, you may define what it means for an axiomatic system to be consistent. You may also define what a proof means. I haven't actually red this, but it may help: https://archive.org/details/IntroductionToTheFoundationsOfMathematics.

let me read it, give me some time please, I appreciate your help. Will be back soon.

 

[Stephen Tashi]

I have learned that some of the basic fundamental, the msot bare bones of mathermatical concepts and definitions have to remain undefined. These are set, element, point, line, proposition, true, false.

What alternatives do we have? As I recall, Aristotle discussed three.

We could have infinite sequences of definitions where each concept is defined in terms of a predecessor concept. Using the notation "A <-- B" for "A is defined interms of B", we could have

A1<--A2<--A3<--A4<--A5<--- etc. forever

Or we could have a circular chain like

A1<--A2<--A3<---A1

Or we could have chain that terminates.

A1<--A2<--A3

(Aristotle discussed such chains in greater generality. For example, in cases where "<--" could mean "is justified by" or "is caused by" or "is proven by".) The oganization of mathematics uses the type of chain that terminates.

.What do I make of this? we have an intuitive guide to all these terms and certainly treat them as we know what they are.

In practice, people treat the terminal element in the chain of definitions in different ways.

The intuitive way to treat it is to assure ourselves that we do know the meaning of undefined terms. We think of the undefined term as denoting something that exists. So "a point" is as real to us as "a coffee cup". We think we know what a coffee cup is, but none of us has memorized a long list of the properties of a coffee cup. We think we can answer questions that crop up about coffee cups without such a list. So if someone asks "Will a coffee cup fit in a suitcase ?" we can answer "yes" without citing a passage in the book that lists "fits in suitcases" as a property of "coffee cup". Likewise, if a person say "Given a circle, is there a point inside it?", we tend to answer "yes" without consulting a list of axioms.

In pure mathematics, the use of intuition hasn't proven completely reliable. So "the game" of pure mathematics is rely only on properties of things that are explicitly stated. Teaching math involves teaching two contradictory techniques. On the one hand, students need to develop intuition. On the other hand they need to learn "the game". Intuitive reasoning is familiar, "The game" is very legalistic and hair splitting. Its crazy to advocate "the game" of axiomatics as only valid approach to life in general, because it is completely impractical to do things like drinking coffee according to a system of definitions and axioms.

Students are accustomed to the self-contradictory aspects of education. Teacher A says to do thing this way, teacher B says to do things another way. They can handle it.

We were discussing this with friends, and I simply stated one of courants views roughly as being the student should not be concerned with the philosophy or nature of mathematical objects, rather it is the relations and operations between them that are more important.

It still blows my mind somewhat.

( You might enjoy investigating the analogous situation in physics. I've read that Leibnitz asserted that space and time were not "things" but only know as relations among things. In "Operationalism", the procedure of measurement is advocated as the basis for definitions of physical concepts: http://plato.stanford.edu/entries/operationalism/ )

 

[Logical Dog]

What alternatives do we have? As I recall, Aristotle discussed three.

We could have infinite sequences of definitions where each concept is defined in terms of a predecessor concept. Using the notation "A <-- B" for "A is defined interms of B", we could have

A1<--A2<--A3<--A4<--A5<--- etc. forever

thank you for you answer..it will take me some time to fully understand it Mr T. I will get back to you.

 

[Logical Dog]

That is right. Then you can take a set of propositions and call it an axiomatic system. In set theory and logic, you may define what it means for an axiomatic system to be consistent. You may also define what a proof means. I haven't actually red this, but it may help: https://archive.org/details/IntroductionToTheFoundationsOfMathematics.

Is this a satisfactory explanation of the axiomatic method?

Proposition: undefined, intuitively can be said to be a statement that can be verified to be either true or false.
Axiom: An initial proposition assumed to be true.

We shall call the problems of the universe which mathematics seek to model and solve as nature. The basic process for trying to develop a mathematical model of nature is this:
1. The mathematician observes some particular property of nature that is of interest.
2. He then develops a set of axioms.
3. He then uses the methods of logical deduction and proof to derive a theorem.
4. He tests this theorem to see if it is an adequate representation of the phenomena in nature (does it model it accurately enough? Does it model outcomes to a satisfactory extent?)
5. If not, he goes back to steps 1 and 2.

How can we define what a proof means? i thought all proofs must be derived through logic and mathematical induction..? or am i confused again :) I don't think I understood that sentence. I know a little bit of logic so you may be able to use a very simple example if you like.

 

[Logical Dog]

What alternatives do we have? As I recall, Aristotle discussed three.

We could have infinite sequences of definitions where each concept is defined in terms of a predecessor concept. Using the notation "A <-- B" for "A is defined interms of B", we could have

A1<--A2<--A3<--A4<--A5<--- etc. forever

Or we could have a circular chain like

A1<--A2<--A3<---A1

Or we could have chain that terminates.

A1<--A2<--A3

(Aristotle discussed such chains in greater generality. For example, in cases where "<--" could mean "is justified by" or "is caused by" or "is proven by".)
( You might enjoy investigating the analogous situation in physics. I've read that Leibnitz asserted that space and time were not "things" but only know as relations among things. In "Operationalism", the procedure of measurement is advocated as the basis for definitions of physical concepts: http://plato.stanford.edu/entries/operationalism/ )

I do not think I can comprehend the last bit about space and time, but I understood the rest.

"Every demonstrable science must begin with indemonstrable principles..."

can you explain operationalism to me if possible ? :)
edit: i think I understand it, but what is the significance of the space and time bit?

 

[Lucas SV]

Proposition: undefined, intuitively can be said to be a statement that can be verified to be either true or false.
Axiom: An initial proposition assumed to be true.

Yes, I would describe a proposition that way. However in the case of axiom, I should say that you don't need the words 'assumed to be true', as I will explain. It is better to define an axiomatic system as a finite collection (whatever finite collection means) of propositions. Then we say a proposition that belongs to that collection is called an axiom.

How can we define what a proof means? i thought all proofs must be derived through logic and mathematical induction..? or am i confused again :) I don't think I understood that sentence. I know a little bit of logic so you may be able to use a very simple example if you like.

Now the reason why we use the wording 'assumed to be true' when referring to axioms is because that is what we do when we prove theorems. If statement $S$ is a theorem in an axiomatic system $\Sigma$, this means that $\Sigma\implies S$. From the way implication works, to prove a theorem it is clear that you must start by assuming $\Sigma$ is true and showing that the truth of $S$ follows from such assumption. $S$ can actually be false if you choose a different axiomatic system. $S$ can also be independent of $\Sigma$. An example is the continuum hypothesis (CH), which is independent of the ZFC axioms.

More technically a proof is a sequence of statements, in which the final statement should be what you are trying to prove. Look up formal proof https://en.wikipedia.org/wiki/Formal_proof. Also so far from what I red in the book, chapter 1, section 4 is the first instance where the author talks about axiomatic systems (He calls them axiom systems).

We shall call the problems of the universe which mathematics seek to model and solve as nature. The basic process for trying to develop a mathematical model of nature is this:
1. The mathematician observes some particular property of nature that is of interest.
2. He then develops a set of axioms.
3. He then uses the methods of logical deduction and proof to derive a theorem.
4. He tests this theorem to see if it is an adequate representation of the phenomena in nature (does it model it accurately enough? Does it model outcomes to a satisfactory extent?)
5. If not, he goes back to steps 1 and 2.

I would add, that before he starts proving theorems, he needs to know what the theorems (or conjectures) are. The idea is that usually a lot is already known about the theory, before it has been axiomatized. Take calculus for example. Computations in calculus could be made and the subject had already a good amount of content, before $\epsilon,\delta$ definitions were created. Mathematics will also test 'theorems', the ones which have not been rigorously proven. But that happens in many cases, before axiomatization.
1. Choose the mathematical field (which is already well established, but its logical foundations are insecure)
2. Choose the set of axioms that best describes the objects studied in the field.
3. Take the basic facts that have been taken for granted in the field, and derive them as theorems from the axioms by using proofs. Do the same for other important propositions in the field.
4. Ask what is the power of the (mathematical, not scientific) theory you have created. How many examples in the field does it encompass. How useful is the theorems you have proven.
5. If your theory does not encompass sufficiently many examples which appear in the field, repeat 2, 3, and so on. If your theory describes the field correctly but is not very useful, repeat 3, 4 and so on.