darrough
Math only requires that it is logically consistent

Could I add ?

Math only requires that it is logically self consistent : it does not need to be consistent with anything else.

BR549
There is no need to disparage a noble profession.

Can't see where you make that out, no one is disparaging anybody.
But Darrough is right, mathematics may have started out as just a tool of someone elese trade, but these days it is a trade in its own right.
I can't agree with WBahn that the differences are unimportant, however.
I see many, students and old hands, here and elsewhere, who are justifiably (IMHO) confused by these differences.
That is why I am asking for, and collecting, examples.

 

"Can't see where you make that out, no one is disparaging anybody."

stu------bless your heart......that was not an accusation.....that was my feeble attempt at humor.

An engineering insider comment. That's all.....sorry bout that.

Whether in 1st grade or the highest level....math is pure.

Math is ordered procedure. It ignores(purity) the result.

The magnitude and complexity of the terms have made it a science.

Results are totality dependent on terms not the procedure.

How can a wrench be wrong?

Defining and relating the terms is the problem.

My personal favorite is point particle.

Some mathematicians and physicists agree on the terms and some don't.

Also the use of a result as a next term.

This is how incomplete or inaccurate terms get amplified.

Context must always accompany results in physics.

Physics failed when it allowed a point for context.

Any theory that uses points for other than a static location, will be an example of what you are looking for.

In math, these equations are valid.

In physics the point is incomplete(because it has no terms) and the equation fails.

Assigning a separate energetic entity that has several maintained and adjustable properties to a point......is the most illegal and cowardly math operation ever preformed.

I shouldn't say that, it wasn't a math operation.....it was a term definition.

But both sides love and embrace it.

That's why to me there is no difference.

 

BR549
My personal favorite is point particle.

I would be grateful if you would amplify your example (I didn't say point )
I am not aware of any conflict in any discipline over this term.

I'm grateful your comment about Engineers was meant in jest.
A suitable smiley helps make that point next time.

WBahn
Separated by a Common Language

A veritably sagacious quote.

However I would say that there is a discernable underlying commonality between the use of the word Field in Maths, Physics and Agriculture and Academia to boot.
That is, it is the stage whereon all the action takes place.

But the point is that not all the action in Agriculture has counterparts in Physics and vice versa.

 

I think my prior point was not fully understood. There is no truth in mathematics. A "true" statement is always a statement about reality. Thus statements in physics are either true or false. Statements in mathematics are not true or false; they are consistent or inconsistent, logical or illogical. Mind blowing, huh? Consider the statement: "this statement is false". The statement can be created in language and be both true and false only because it does not refer to reality. If a statement refers to reality then it has to be true or false.

 

I think my prior point was not fully understood. There is no truth in mathematics. A "true" statement is always a statement about reality. Thus statements in physics are either true or false. Statements in mathematics are not true or false; they are consistent or inconsistent, logical or illogical. Mind blowing, huh? Consider the statement: "this statement is false". The statement can be created in language and be both true and false only because it does not refer to reality. If a statement refers to reality then it has to be true or false.

There IS such a thing called mathematical reality... and the debate isn't over yet.

 

Hello Darrough. What does your cat think of Schrodinger and your statement that everything in reality is true or false?

Actually, we accepts axioms as tue in mathematics. They don't have to be consistent, though it helps.

 

studiot,

"I would be grateful if you would amplify your example (I didn't say point)".

What aspect of the example may I try to clarify. I am also not sure what "I didn't say point" means.

"I am not aware of any conflict in any discipline over this term."

There is conflict of using a point particle or a point as an real entity and emission source(or any kind of source).

How can a dimensionless point cause a action or property?

See the conflict?

 

There IS such a thing called mathematical reality... and the debate isn't over yet.

There are a number of physicists that take that position. And to be fair, when I said "There is no truth in mathematics" I should have said there is no factual truth in mathematics. Truth in mathematics is defined as being logically possible.

If you would like to elaborate on mathematical realities I would like to know more about it.

 

Hello Darrough. What does your cat think of Schrodinger and your statement that everything in reality is true or false?

I have never taken a course in modern physics, but I did listen to an audio book. (It is as mind blowing as math ever was.) It may very well be that propositions about (physical) reality do not have single truth states. I do not think that is a foregone conclusion though. As I understand it, there are different "interpretations" for example "copenhagen" and "many worlds". Doesn't "many worlds" hold that there are an infinite number of realities, but in each one a proposition about (physical) reality has single truth state?

Actually, we accepts axioms as true in mathematics. They don't have to be consistent, though it helps.

I know there are some new approaches to logic, such as quantum logic. At least within the framework of classical logic, truth in mathematics is analytic truth whereas truth in physical science is synthetic or empirical or factual truth. As far as axioms not having to be consistent, I have never heard of that. Please elaborate.

 

As far as axioms not having to be consistent, I have never heard of that. Please elaborate.

An axiom is basically nothing more than a starting point that can't be proven (at least not within the system it describes) and must simply be accepted. For this reason, they must be (or should be) "obviously true".

Anything can be an axiom provided it can't be proven (or disproven) within the system (otherwise it would not be an axiom at all, but rather a theorem). But what if two axioms are contradictory? Well, you have some choices to make. You can throw one of them out (but which one?), you can create special rules to resolve the difficulty, or you can stick to your guns and basically say that they aren't contradictory and that the system just has to be narrowed accordingly.

For instance, in arithmetic the commutative property of addition, a+b=b+a is an axiom. It can't be proven. We could declare a commutative property of subtraction, namely that a-b=b-a, but we would run into problems. So we would throw it out. But we could simply declare that subtraction only applied to instances where a=b. It would be rather stupid to do so, but it would be a way of resolving the problem. You might be tempted to say that we can't do that because the operation has to be defined for all elements of the number system, to which I would point out that we already take this route; consider division -- a/b is only defined for b not equal to zero. Why? Because it raises inconvenient issues if we don't adopt that special rule.

 

BR549
What aspect of the example may I try to clarify. I am also not sure what "I didn't say point" means.

I am saying that I don't understand this diatribe against the point particle. You misunderstand the term.

The other statement was a joke about the fact that I did not make a joke (pun) about the word point.
I thought we just agreed in an earlier thread that we both like jokes?
Sorry if this one fell flat.

My personal favorite is point particle.
Some mathematicians and physicists agree on the terms and some don't.
Also the use of a result as a next term.
This is how incomplete or inaccurate terms get amplified.
Context must always accompany results in physics.
Physics failed when it allowed a point for context.
Any theory that uses points for other than a static location, will be an example of what you are looking for.
In math, these equations are valid.
In physics the point is incomplete(because it has no terms) and the equation fails.
Assigning a separate energetic entity that has several maintained and adjustable properties to a point......is the most illegal and cowardly math operation ever preformed.
I shouldn't say that, it wasn't a math operation.....it was a term definition.
But both sides love and embrace it.
That's why to me there is no difference.

The term 'point particle' originated in classical mechanics and is simply a rigid body whose dimensions are insignificant compared to the system under consideration.
So we consider the Earth as a 'point particle' when discussing the orbital mechanics around the Sun.
Whilst I expect one could find an equation that misused the concept if one looked hard enough, it is not true to say that equations fail.
For instance the balancing of moments about a point leads to comfirmable equations and measurements of real world forces and perhaps energies.

 

darrough
As far as axioms not having to be consistent, I have never heard of that. Please elaborate.

A system of mathematics is based on a set of axioms.

A system based on a singleton axiom, is based on a statement that has nothing to be consistent with.

Take for instance the axiom

A line has at most two ends. (note I have not restricted my line to being straight on crooked or anything)

A suprising amount of mathematical system can be deduced from this one axiom.

Theorem: A line with zero ends contains at least one loop or extends to infinity.

Theorem: A line with one end has at least one loop.

Theorem: A line with two ends is bounded.

And more.

 

A system of mathematics is based on a set of axioms.

A system based on a singleton axiom, is based on a statement that has nothing to be consistent with.

Take for instance the axiom

A line has at most two ends. (note I have not restricted my line to being straight on crooked or anything)

A suprising amount of mathematical system can be deduced from this one axiom.

Theorem: A line with zero ends contains at least one loop or extends to infinity.

Theorem: A line with one end has at least one loop.

Theorem: A line with two ends is bounded.

And more.

But if a line with zero ends is allowed to be one that extends to infinity (presumably in both directions), then why does a line with one end have to have at least one loop? Why can't it have one end that extends to infinity?

I realize that this is just a thrown together example (isn't it?).

To extend the discussion: We could have started with the axiom that a line has exactly two ends. In doing so, we would be defining a different set of mathematics in which "lines" that contain loops or extend to infinity are simply not lines at all, but are called something else.

 

Yes there's lots of possibilities for development of the calculus or algebra of lines.

 

An axiom is basically nothing more than a starting point that can't be proven (at least not within the system it describes) and must simply be accepted. For this reason, they must be (or should be) "obviously true".

And axioms are not all of an exclusively mathematical nature, such as:

• I think, therefore I exist
• There is a God
• Solipsism is wrong

Of course, a discussion like this would probably belong to a different kind of forum, and it would also be an endless debate (see the philosophical arguments by Mr David Hume). But my point is that the existential implications of mathematics are an unavoidable thing.
Are mathematics discovered or created? Would an argument on this question be useful or moot?

 

And axioms are not all of an exclusively mathematical nature, such as:

• I think, therefore I exist
• There is a God
• Solipsism is wrong

Of course, a discussion like this would probably belong to a different kind of forum, and it would also be an endless debate (see the philosophical arguments by Mr David Hume). But my point is that the existential implications of mathematics are an unavoidable thing.
Are mathematics discovered or created? Would an argument on this question be useful or moot?

Mathematical and physical axioms are almost always qualified with being something that is "obviously" true and that, therefore, is not a contentious issue in and of itself. When you say that addition is commutative, no one jumps up and down claiming that it isn't and no one questions the validity of the resulting mathematical system based on challenging the axioms upon which it is founded. That's not to say that they couldn't, but it would be a big, uphill battle for them.

But when you build your system on an axiom that is not accepted as "obvious" on a nearly universal basis, you are asking for nothing but problems from the start. This is especially true if you then proceed to justify the validity of the axioms using the system that was built upon them. I've heard people assert that the Bible is absolutely true because it is the Word of God, but then assert that God exists because the Bible says so.

Similarly, you have to be careful with an axiom like your first one. What does it mean to exist? If thinking is sufficient for existence, is it also necessary?

As for whether mathematics is discovered or created, that would definitely fall in the category of bad axioms because there is anything but obvious and near universal agreement on that point -- there isn't even agreement on what a rigorous distinction between "discovered" and "created" would be or just what those two words mean to begin with.

 

OK well then we will agree to disagree. I guess I would say you don't get mathematical logic or formal systems.

Mathematicians don't think of axioms as "contentious" or "obvious". They pick an axiom because they like the theorems that they can prove with the axiom. They feel free to pick any axiom provided it is not contradictory to axioms that they have already chosen. In fact quite often they will develop a branch of math using a certain axiom and another using the negation of the axiom and consider both branches to be good, valid mathematics. This will not bother them in the least. They will consider both to be "true". That's because true means something different in mathematics. It means that the statement is logically consistent within the system.

 

Well, the very word "axiom" comes from Greek and basically means "that which is self-evident".

That's not to say that you can't declare whatever you want as your system's axioms and then proceed to live with them.

 

OK well then we will agree to disagree. I guess I would say you don't get mathematical logic or formal systems.

Mathematicians don't think of axioms as "contentious" or "obvious". They pick an axiom because they like the theorems that they can prove with the axiom. They feel free to pick any axiom provided it is not contradictory to axioms that they have already chosen. In fact quite often they will develop a branch of math using a certain axiom and another using the negation of the axiom and consider both branches to be good, valid mathematics. This will not bother them in the least. They will consider both to be "true". That's because true means something different in mathematics. It means that the statement is logically consistent within the system.

But I do get mathematical logic and formal systems (or at least I think I do ).

The point I'm trying to make is more or less like Wbahn said: "... you have to be careful with an axiom like your first one. What does it mean to exist? If thinking is sufficient for existence, is it also necessary?" (and BTW, I didn't mean to bring religion into the subject, but rather philosophy)
The way I understand axioms is that they constitute the basic building blocks from which the rest of all logic is developed within a system. But when the system is existence itself (the universe and all that lies within) then philosophical, and even theological arguments are unavoidable... tell me, how far do you think Gödel's incompleteness theorem reaches? Is his "discovery" limited to the realm of mathematics only?

EDIT: just to be clear, although I'm only an engineer, I've always been interested in subjects that relate directly (or even indirectly) to my profession. Several years ago I read G.H. Hardy's A Mathematician's Apology, where he explained to the uninitiated the concept of mathematical proof. I know that some members of this forum don't have the best of opinions regarding Mr Hardy, and that having read one book on the subject does not make me an expert. But I'm sure that everyone agrees that that little publication of his is a valuable tool for anyone who's trying to understand the formal concept of mathematical proof.

 

As ever, life is more complicated.
The Ancient Greeks, didn't use the word axiom, they use the word postulate.
The most complete and thorough example of a full blown axiomatic system of mathematics, not some hastily thrown together one like my example, was is of course, Euclids elements.

But the five postulates (axioms) of Euclids elements do not stand alone.

There are five more that are called common notions and twenty three more statements acting in a supporting role called definitions.

http://aleph0.clarku.edu/~djoyce/elements/bookI/bookI.html

Where the axioms finish and other forms of statement accepted as 'true' take over is a grey area.

Then we fast forward two and a half thousand years to Godel and his incompleteness theorem.

Darrough I gave you a simple axiomatic system where the axiom does not have to be consistent with anything, although obviously deduced theorems have to be consistent with it, and has WBahn, showed, with each other.
But you did not comment.
You might like to add your thoughts on Godel into this as well?