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what's the role of axioms within a mathematical theory ?
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Philip Lee
Aug 25, 2017, 5:38:45 AM8/25/17
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Is a set of axioms the starting point of any math theory, and upon which a theory could be built up ? If not, what's the role of axioms within a math theory ?
FromTheRafters
Aug 25, 2017, 6:28:46 AM8/25/17
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Philip Lee pretended :
a big if...then... statement. If you accept this this and this axiom,
then it probably follows that this and that 'conjecture' is true. One
then goes on to use the axioms followers have agreed to be true to
prove that the conjecture is true in this system by using the axioms in
a proof. Once there is a proof that is accepted, the conjecture becomes
a theorem.
Theorems and axioms can then be used to prove other conjectures. Some
theorems, especially small ones, are used to prove larger conjectures
and get called lemmas.
Axioms are the fundamental building blocks which everything else in the
system depends upon.
> Is a set of axioms the starting point of any math theory, and upon which a
> theory could be built up ? If not, what's the role of axioms within a math
> theory ?
Axioms are like assumptions which are stipulated to be true. It is like
> theory could be built up ? If not, what's the role of axioms within a math
> theory ?
a big if...then... statement. If you accept this this and this axiom,
then it probably follows that this and that 'conjecture' is true. One
then goes on to use the axioms followers have agreed to be true to
prove that the conjecture is true in this system by using the axioms in
a proof. Once there is a proof that is accepted, the conjecture becomes
a theorem.
Theorems and axioms can then be used to prove other conjectures. Some
theorems, especially small ones, are used to prove larger conjectures
and get called lemmas.
Axioms are the fundamental building blocks which everything else in the
system depends upon.
Peter Percival
Aug 25, 2017, 6:32:45 AM8/25/17
to
true and if the underlying logic is sound, then the theorems are also true.
One point of view--perhaps an old fashioned one--is that the axioms are
self-evidently true and thus the truth of the theorems is guaranteed.
One problem is that if (say) the axioms of Euclidean geometry are true
and the theorems of Euclidean geometry are indisputable facts, then why
study any other geometry? You may substitute the name of your favourite
geometry for "Euclidean".
Another point of view is that the axioms encapsulate a few principles of
interest and their consequences are studied for their own sake. Vulgar
matters like truth don't come in to it.
In any case, meta-theorems are often at least as interesting as
theorems. For example, if such-and-such axioms are consistent they
remain so when axiom so-and-so is added. I would imagine (I confess,
I'm guessing) that meta-mathematics would be impossible outside of the
axiomatic approach. In logic it is certainly the case that the
meta-theorems are vastly more interesting than the theorems.
Whatever your view of axioms, note that theorems come first (I mean
first historically, not logically). You cannot know what axioms to
adopt before you know what theorems you want to prove.
Mention of "true" above may cause you to ask "true of what?" I have no
idea.
>
--
Do, as a concession to my poor wits, Lord Darlington, just explain
to me what you really mean.
I think I had better not, Duchess. Nowadays to be intelligible is
to be found out. -- Oscar Wilde, Lady Windermere's Fan
FredJeffries
Aug 25, 2017, 10:57:45 AM8/25/17
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On Friday, August 25, 2017 at 2:38:45 AM UTC-7, Philip Lee wrote:
> Is a set of axioms the starting point of any math theory, and upon which a theory could be built up ? If not, what's the role of axioms within a math theory ?
It's a tool which (among other things) helps you uncover hidden assumptions.
> Is a set of axioms the starting point of any math theory, and upon which a theory could be built up ? If not, what's the role of axioms within a math theory ?
Philip Lee
Aug 25, 2017, 11:59:01 AM8/25/17
to
According to Wikipedia https://en.wikipedia.org/wiki/Axiom
[In the modern understanding, it is not correct to say that the axioms of field theory are "propositions that are regarded as true without proof." Rather, the field axioms are a set of constraints.]
Now I have another question to ask: Does every mathematical theory need to be axiomatized[https://en.wikipedia.org/wiki/Axiomatic_system#Axiomatization]?
If yes, why?
If no, in what cases a math theory need to be axiomatized?
A mathematical theory here doesn't include the axiomatic system, More details about what I mean , please see here[https://en.wikipedia.org/wiki/Axiomatic_system#History].
Arturo Magidin
Aug 25, 2017, 12:33:46 PM8/25/17
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On Friday, August 25, 2017 at 4:38:45 AM UTC-5, Philip Lee wrote:
> Is a set of axioms the starting point of any math theory, and upon which a theory could be built up ? If not, what's the role of axioms within a math theory ?
Yes. In formal axiomatic theory, the set of axioms represent the "rules of the game", the basic assumptions on which everything rests. They need not have meaning, they need not represent anything, they are just formal rules.
> Is a set of axioms the starting point of any math theory, and upon which a theory could be built up ? If not, what's the role of axioms within a math theory ?
In practice, axioms play the role of the basic assumptions. While in ancient greek they were thought to be "obviously true and so not in need of proof", these days we have abandoned the idea of "true". They simply are. We often state them so as to model specific situations we are interested in.
--
Arturo Magidin
Philip Lee
Aug 25, 2017, 1:10:03 PM8/25/17
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On Saturday, August 26, 2017 at 12:33:46 AM UTC+8, Arturo Magidin wrote:
> On Friday, August 25, 2017 at 4:38:45 AM UTC-5, Philip Lee wrote:
> > Is a set of axioms the starting point of any math theory, and upon which a theory could be built up ? If not, what's the role of axioms within a math theory ?
>
> Yes. In formal axiomatic theory, the set of axioms represent the "rules of the game", the basic assumptions on which everything rests.
>
As I have learnt from Wikipedia [https://en.wikipedia.org/wiki/Axiomatic_system#Issues]
> On Friday, August 25, 2017 at 4:38:45 AM UTC-5, Philip Lee wrote:
> > Is a set of axioms the starting point of any math theory, and upon which a theory could be built up ? If not, what's the role of axioms within a math theory ?
>
> Yes. In formal axiomatic theory, the set of axioms represent the "rules of the game", the basic assumptions on which everything rests.
>
"Not every consistent body of propositions can be captured by a describable collection of axioms."
[https://en.wikipedia.org/wiki/Axiom#Examples_2]
"Combinatorics is an example of a field of mathematics which does not, in general, follow the axiomatic method."
Does this mean Combinatorics is not based on axioms?
Arturo Magidin
Aug 25, 2017, 2:01:02 PM8/25/17
to
On Friday, August 25, 2017 at 12:10:03 PM UTC-5, Philip Lee wrote:
> On Saturday, August 26, 2017 at 12:33:46 AM UTC+8, Arturo Magidin wrote:
> > On Friday, August 25, 2017 at 4:38:45 AM UTC-5, Philip Lee wrote:
> > > Is a set of axioms the starting point of any math theory, and upon which a theory could be built up ? If not, what's the role of axioms within a math theory ?
> >
> > Yes. In formal axiomatic theory, the set of axioms represent the "rules of the game", the basic assumptions on which everything rests.
> >
> As I have learnt from Wikipedia [https://en.wikipedia.org/wiki/Axiomatic_system#Issues]
> "Not every consistent body of propositions can be captured by a describable collection of axioms."
> [https://en.wikipedia.org/wiki/Axiom#Examples_2]
This is hopelessly vague and imprecise, since what is and what is not "describable" is vague and imprecise in this statement. That said, this is neither here nor there to the original question. This is about what axiomatic systems can or cannot achieve, not about what axioms are.
> On Saturday, August 26, 2017 at 12:33:46 AM UTC+8, Arturo Magidin wrote:
> > On Friday, August 25, 2017 at 4:38:45 AM UTC-5, Philip Lee wrote:
> > > Is a set of axioms the starting point of any math theory, and upon which a theory could be built up ? If not, what's the role of axioms within a math theory ?
> >
> > Yes. In formal axiomatic theory, the set of axioms represent the "rules of the game", the basic assumptions on which everything rests.
> >
> As I have learnt from Wikipedia [https://en.wikipedia.org/wiki/Axiomatic_system#Issues]
> "Not every consistent body of propositions can be captured by a describable collection of axioms."
> [https://en.wikipedia.org/wiki/Axiom#Examples_2]
> "Combinatorics is an example of a field of mathematics which does not, in general, follow the axiomatic method."
>
> Does this mean Combinatorics is not based on axioms?
On the other hand, combinatorics generally follows a certain set of rules or axioms, such as the Sum Rule ("Suppose A and B are independent events; if event A can occur in n ways, and event B can occur in m ways, then the number of ways in which either event A or B can occur is n+m ways") and the Product Rule ("Suppose A and B are independent events; if event A can occur in n ways, and event B can occur in m ways, then the number of ways in which both events A and B can occur is nm.") One could argue that these basic rules can play the role of axioms, but since we don't have a set of primitive notions and a clear list of axioms, Combinatorics is not developed using the axiomatic method.
Note: "axiomatic method" =/= "based on axioms". The axiomatic method requires that you be based on axioms, but there are a bunch of other requirements as well.
--
Arturo Magidin
Bill
Aug 25, 2017, 8:26:37 PM8/25/17
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Philip Lee wrote:
> Is a set of axioms the starting point of any math theory, and upon which a theory could be built up ? If not, what's the role of axioms within a math theory ?
What "math theory" are you referring to? What do you have in mind?
> Is a set of axioms the starting point of any math theory, and upon which a theory could be built up ? If not, what's the role of axioms within a math theory ?
Philip Lee
Aug 26, 2017, 1:53:39 AM8/26/17
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Philip Lee
Aug 28, 2017, 12:49:02 PM8/28/17
to
On Saturday, August 26, 2017 at 2:01:02 AM UTC+8, Arturo Magidin wrote:
> On Friday, August 25, 2017 at 12:10:03 PM UTC-5, Philip Lee wrote:
> > On Saturday, August 26, 2017 at 12:33:46 AM UTC+8, Arturo Magidin wrote:
> > > On Friday, August 25, 2017 at 4:38:45 AM UTC-5, Philip Lee wrote:
> > > > Is a set of axioms the starting point of any math theory, and upon which a theory could be built up ? If not, what's the role of axioms within a math theory ?
> > >
> > > Yes. In formal axiomatic theory, the set of axioms represent the "rules of the game", the basic assumptions on which everything rests.
> > >
> > As I have learnt from Wikipedia [https://en.wikipedia.org/wiki/Axiomatic_system#Issues]
> > "Not every consistent body of propositions can be captured by a describable collection of axioms."
> > [https://en.wikipedia.org/wiki/Axiom#Examples_2]
>
> This is hopelessly vague and imprecise, since what is and what is not "describable" is vague and imprecise in this statement. That said, this is neither here nor there to the original question. This is about what axiomatic systems can or cannot achieve, not about what axioms are.
>
1. You said ''' the set of axioms represent the "rules of the game", the basic assumptions on which **everything** rests''', so I quoted "**Not every** consistent body of propositions can be captured by a describable collection of axioms." as a contrary to tell there are limitations on the the axiomatic method .
> On Friday, August 25, 2017 at 12:10:03 PM UTC-5, Philip Lee wrote:
> > On Saturday, August 26, 2017 at 12:33:46 AM UTC+8, Arturo Magidin wrote:
> > > On Friday, August 25, 2017 at 4:38:45 AM UTC-5, Philip Lee wrote:
> > > > Is a set of axioms the starting point of any math theory, and upon which a theory could be built up ? If not, what's the role of axioms within a math theory ?
> > >
> > > Yes. In formal axiomatic theory, the set of axioms represent the "rules of the game", the basic assumptions on which everything rests.
> > >
> > As I have learnt from Wikipedia [https://en.wikipedia.org/wiki/Axiomatic_system#Issues]
> > "Not every consistent body of propositions can be captured by a describable collection of axioms."
> > [https://en.wikipedia.org/wiki/Axiom#Examples_2]
>
> This is hopelessly vague and imprecise, since what is and what is not "describable" is vague and imprecise in this statement. That said, this is neither here nor there to the original question. This is about what axiomatic systems can or cannot achieve, not about what axioms are.
>
2. If axioms were not used in axiomatic method , what are the other usages of them?
>
> > "Combinatorics is an example of a field of mathematics which does not, in general, follow the axiomatic method."
> >
> > Does this mean Combinatorics is not based on axioms?
>
> As opposed to areas like Analysis, Abstract Algebra, etc., combinatorics does not set forth a set of primitive notions and a set of axioms that they follow. So they do not follow the axiomatic method.
>
> On the other hand, combinatorics generally follows a certain set of rules or axioms, such as the Sum Rule ("Suppose A and B are independent events; if event A can occur in n ways, and event B can occur in m ways, then the number of ways in which either event A or B can occur is n+m ways") and the Product Rule ("Suppose A and B are independent events; if event A can occur in n ways, and event B can occur in m ways, then the number of ways in which both events A and B can occur is nm.") One could argue that these basic rules can play the role of axioms, but since we don't have a set of primitive notions and a clear list of axioms, Combinatorics is not developed using the axiomatic method.
>
> Note: "axiomatic method" =/= "based on axioms". The axiomatic method requires that you be based on axioms, but there are a bunch of other requirements as well.
>
3. "The axiomatic method requires that you be based on axioms, but there are a bunch of other requirements as well. " What are the other requirements ? I cannot find these requirements by referring to the definition of axiomatic method. As for primitives, I think they are part of the axioms according to the representation I learnt from https://www.britannica.com/topic/axiomatic-method
> > "Combinatorics is an example of a field of mathematics which does not, in general, follow the axiomatic method."
> >
> > Does this mean Combinatorics is not based on axioms?
>
> As opposed to areas like Analysis, Abstract Algebra, etc., combinatorics does not set forth a set of primitive notions and a set of axioms that they follow. So they do not follow the axiomatic method.
>
> On the other hand, combinatorics generally follows a certain set of rules or axioms, such as the Sum Rule ("Suppose A and B are independent events; if event A can occur in n ways, and event B can occur in m ways, then the number of ways in which either event A or B can occur is n+m ways") and the Product Rule ("Suppose A and B are independent events; if event A can occur in n ways, and event B can occur in m ways, then the number of ways in which both events A and B can occur is nm.") One could argue that these basic rules can play the role of axioms, but since we don't have a set of primitive notions and a clear list of axioms, Combinatorics is not developed using the axiomatic method.
>
> Note: "axiomatic method" =/= "based on axioms". The axiomatic method requires that you be based on axioms, but there are a bunch of other requirements as well.
>
> --
> Arturo Magidin
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