Is math the structure of the Universe?

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Jim_Baur said:
Is math the fundamental structure of the universe?
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No. Mathematics is an axiomatic system, so the results you get out depend on the axioms you put in. For example, depending on what you choose for the Parallel Axiom, you can get Spherical, Euclidian or Lobachevskian geometry. Each will give you different answers to some questions. The angles of a triangle add up to less than 180° in Lobachevskian geometry, exactly 180° in Euclidian geometry and more than 180° in Spherical geometry.

Change the axioms and you change the mathematics.

rossum
 
I think it would be more proper to say that math describes (or attempts to) describe the structure of the universe. New mathematical methods/systems are “invented” for the purpose of describing observed phenomena.
 
Some mathematicians get misty eyed about this. I’d say it’s probably true that everything can ultimately be described by math, but not that math is the basis of reality as, for instance, it’s implausible that every molecule in planet earth is continuously calculating its motion around the sun.
 
The fact that the universe can be described by math points to order in the natural world. I do think though that math only describes a subset of the physical universe.

For instance isn’t it remarkable that many physical systems behave according to conic sections?
 
I never understood maths. What is 1, and what is 2, in terms of a universe that slips through your fingers the closer you focus in on any particular point.
 
How does this effect science?

Is reason the structure?

Are the laws?

Do we even know the structure?
 
Paddy_Walker said:
I never understood maths. What is 1, and what is 2, in terms of a universe that slips through your fingers the closer you focus in on any particular point.
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The integers are defined by the Peano Axioms.

Basically, 1 is successor(0) and 2 is successor(1). With a different set of axioms, the numbers would have different properties, for example modular arithmetic: 5 + 6 = 2 (mod 9).

Any mathematical structure is defined by its axioms. Those axioms may, or may not, have some basis in reality.

rossum
 
Jim_Baur said:
How does this effect science?

Is reason the structure?

Are the laws?

Do we even know the structure?
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Math is one of the isomorphic systems that we have for describing/modeling systems. It’s not the only one, but seems to be the most generic one which allows it to be adapted to many problem sets.

“Laws” are descriptions of our observations, not prescriptions.

We’ve got provisional models for various perspectives of the universe. Key word here in “provisional.” Let’s say I displayed for you a sidereal clock (we usually use clocks based on solar time) and let you observe but not take it apart. You may be able to create various models on it’s behaviour ranging equations to predicting what reading will show on the clock to making something that behaves the same but may have different inner workings (since you never got to see them). Despite knowledge of it’s internal structure you are able to make predictions on it’s behaviour and model it.

The reason for this metaphor is that at various scopes we have a limited perspective of the universe (at the astronomical scale, subatomic, and some others). Be the models ever so imperfect many of the models have shown to be of high utility value and have improved over time.
 
Good stuff!

I get the impression that many scientists believe that the universe is powered by math.

I am not saying that they thought about it, but a lot of people that love math and science have never taken the time to think about it since they were in school.

I love the beauty of math and science.

I cannot image life without them.

I do not think that they should impose a limit on their disciplines.

When trying to solve the mysteries of the origins of life and the universe, human reason must use all of its resources–logic, common sense, poetry,all of the arts, philosophy, nature theology, religions and theology.

Good problem solving skills demand that one make a comprehensive list of what we do not know. One these issues, the list is endless.

We have to say: do not tell me you know so much, when there is so much we do not know. This is not to you or any one person; it is just a saying. The saying is trying to say: Be careful.
 
rossum said:
The integers are defined by the Peano Axioms.

Basically, 1 is successor(0) and 2 is successor(1). With a different set of axioms, the numbers would have different properties, for example modular arithmetic: 5 + 6 = 2 (mod 9).

Any mathematical structure is defined by its axioms. Those axioms may, or may not, have some basis in reality.

rossum
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are there an infinite set of numbers between 1 and 2 and 0 and 1 ?
 
Paddy_Walker said:
are there an infinite set of numbers between 1 and 2 and 0 and 1 ?
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What type of numbers are you talking about? Integers, no. Reals, yes. Complex, yes or not defined, depending on how you define “between”.

The answer also depends on which infinity you are talking about. There is a countable infinity of rational numbers between 0 and 1, but not an uncountable infinity. There is an uncountable infinity of real numbers between 0 and 1. And that is before we get into the Aleph numbers.

Yes, I spent five years teaching Mathematics to 11 to 18 year olds. 🙂

rossum
 
rossum said:
What type of numbers are you talking about? Integers, no. Reals, yes. Complex, yes or not defined, depending on how you define “between”.

The answer also depends on which infinity you are talking about. There is a countable infinity of rational numbers between 0 and 1, but not an uncountable infinity. There is an uncountable infinity of real numbers between 0 and 1. And that is before we get into the Aleph numbers.

Yes, I spent five years teaching Mathematics to 11 to 18 year olds. 🙂

rossum
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:hypno:

can i start with something easier; what is 1? philosophically
 
LEMAITRE said:
Fibonacci, natures number system. What does that mean?
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It is an over-exaggeration of finding the Fibonacci sequence (a recursive relationship where the next number is the sum of the previous two, i.e. 1, 1, 2, 3, 5, 8, 13…) in couple places in the world naturally. The series is not nature’s number system.
 
Paddy_Walker said:
can i start with something easier; what is 1? philosophically
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In university, the joke is that you spend the first year of Pure Mathematics learning what 1 is. You spend the second year learning to count from 1 to 2. The rest is trivially derived from those two.

The empty set, {}, is the set which does not contain anything.

Zero is that Cardinal number of the empty set: card({}).

One is the cardinal number of the set containing the empty set: card({{}}).

Clear?

rossum
 
rossum said:
In university, the joke is that you spend the first year of Pure Mathematics learning what 1 is. You spend the second year learning to count from 1 to 2. The rest is trivially derived from those two.

The empty set, {}, is the set which does not contain anything.

Zero is that Cardinal number of the empty set: card({}).

One is the cardinal number of the set containing the empty set: card({{}}).

Clear?

rossum
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yes, clear. 1 doesn’t really exist, its an empty space containing an empty space. so 1 is really nothing. i am to think of numbers as russian dolls of nothingness.
 
LEMAITRE said:
Fibonacci, natures number system. What does that mean?
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If you divide a number in the Fibonacci sequence by the previous number you get 1/1, 2/1, 3/2, 5/3, 8/5, 13/8, 21/13, 34/21, etc.

At the limit, this homes in on the Golden Ratio = 1.6180339…, which is used by artists and architects to produce pleasing proportions. It is also found in natural specimens we find beautiful, so maybe that’s what “natures number system” means.
 
Paddy_Walker said:
yes, clear. 1 doesn’t really exist, its an empty space containing an empty space. so 1 is really nothing. i am to think of numbers as russian dolls of nothingness.
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It kind of sounds like you are looking for a Platonic form.

As used in every day language numbers are an abstraction and an inference preserving metaphor. As abstractions go while you can find physical instantiations of something that shares the properties of the metaphor, but there isn’t instance of the pure abstraction. Trying to treat the abstraction as something concrete is called reification. Note that this is also a classification of logical fallacies when used in logical arguments.

Numbers are useful for constructing logical isomorphism of many systems. But how the numbers map to the system may vary. Operations performed on numbers (multiplication, division, subtraction, addition, and others) often (but not always) can be mapped to physical actions. Whether or not a mapping can be made will depend on the system.

For example, when dealing with sets the + in the expression 1+1=2 would represent the combining of sets and the digits represent set sizes. But when dealing with boolean logic the expression in the expression 1+1=1 the digits represent a bivalent truth value(true/false, yes/no, or many others; the metaphor is multi-layered) and the + operator represents something different.
 
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