Why does math work in our reality?

  • #176
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sorry about that very short post. After I've done the cooking I'll come back and work out exactly what I do think.


If there are no true contradiction, then what can follow is that there is no possible world with true contradiction. Since our world is a possible world, then it follows that there is no true contradiction in our world. Thus, Our world is logically possible.
 
  • #177
apeiron
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If there are no true contradiction, then what can follow is that there is no possible world with true contradiction. Since our world is a possible world, then it follows that there is no true contradiction in our world. Thus, Our world is logically possible.
Alternatively, only "logical contradictions" make reality logically possible.

So we must have both substance and form, chance and necessity, change and stasis, discrete and continuous, atom and void, matter and mind, etc, etc. Each arises as the negation of the other - thesis and antithesis. And then in interaction they create (or in modelling recreate) our reality.

At a superficial level, contradiction seems a bad thing logically. But at a deep level, it is what philosophically and mathematically we have always found.
 
  • #178
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Alternatively, only "logical contradictions" make reality logically possible.
I don` t see how our world is depended on a logical contradiction.


So we must have both substance and form, chance and necessity, change and stasis, discrete and continuous, atom and void, matter and mind, etc, etc. Each arises as the negation of the other - thesis and antithesis. And then in interaction they create (or in modelling recreate) our reality.

At a superficial level, contradiction seems a bad thing logically. But at a deep level, it is what philosophically and mathematically we have always found.

This is rather confusing. What i mean be a "logically possible world" should be interpreted as an semantic for modal logic.
 
  • #179
Pythagorean
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Interesting idea about the brain. Do you think that if viewed as a physical object rather than experiential (if that is a word) the formation of mathematical ideas in it means anything about Nature?
I think, that as we've already generally accepted, it shows us that we can only form a close approximation of nature. Even with our empirical (observation) and mathematical (theory) cleverness, nature is just outside of our perception in the context of science and logic. Traditionally, science and logic required that things were deterministic. But part of accepting that we can't really know things exactly as they makes us give up "confidence" in our deterministic principles, so we have stochastic systems, which is a way to make up for the flaws in understanding that cripple determinism. I would venture to say that stochastic studies are still very much deterministic in the sense that you're still trying to determine things in a logical, rational fashion. You've only relieved yourself of the accountability of being wrong by using words like "confidence" and "probability".

An interesting historical aside is that Riemann seems to think that idea formation was the true model of the universe and tried to compare finite objects to ideas and continuous space to the mind as a whole. His model for the universe was that it had the same processes as the mind.
I thought Einstein had used Riemann Geometry anyway, but I've mostly only heard about GR "in the halls".

I think this philosophical attitude inspired his invention of shocks in non-linear wave equations. The shock is like a "new idea" that lawfully arises in the "mind" to resolve an apparent contradiction, in this case a multi-valued signal. The law is preserved - but the mind/Nature if you will - creates a new object in order to preserve it and also changes the meaning of what a solution to the equation is.

He came up with this idea rather than introducing diffusion terms (changing the law) to prevent a multiple signal. His view I guess was that Nature does not change its laws but rather creates new things if it has to to preserve the laws.

More generally I think that the attitude that unchanging truth is to be discovered behind inexact observation is a guiding koan of modern physics. It can be seen in Einstein,Lorenz, Gallileo, Riemann. This is one reason that I can not accept the view that math is mere modeling. If that were Kepler and Newton's view then the Ptolemaic system would never have been rejected. It was an incredibly accurate model and could be adjusted periodically to preserve its accuracy. It was rejected precisely because it did not present universal law - in fact it contradicted the idea of universal law since it had to be modified from time to time - sort of like putting diffusion terms into the wave equation.
But to me, this says that physics is not mere modeling. If math was all that mattered, than the Ptolemaic system would have been fine. Observations matter too, though. There's something just weird about planets doing little loop-de-loops out of nowhere. That would be unsettling to must of us who were genuinely curious and studied the night sky.

Just like, for Einstein, it would have been unsettling if gravity was instantaneous and that.. if the Sun disappeared, we'd still stay forever bathed in it's light as the gravity instantly let us go, but the light took time traveling the speed C to us.

Math is not the one to go to to say "this is weird... there's something not right about this", intuition is. We then go and break apart our question into mathematics to answer the question. That's what math does for us: makes the information manageable.

Realize that this is also a matter of logistics. Nobody wants to have to adjust their calendar, it's inconvenient. And then there's the whole egocentric part about being at the center of the universe....

This thread because of its empiricist/positivist bias doesn't even care about this and doesn't care what thoughts inspired the progress of science. In fact I am sure that someone in this thread will write that the Ptolemaic system was perfectly fine and Gallileo and Kepler and Newton and Riemann and Einstein and all of the others were jerks - they just didn't understand anything.
I disagree. Thoughts that inspired the progress of science are very fascinating to most of us, especially when you realize how often mistakes and random guessing have helped to discover nature. We love seeing where intuition helps and where it hinders. No rational individual is going to say Ptolemaic system was fine simply because there's no reason to believe that we'd be the center of the universe and there's a simpler explanation.
 
  • #180
SixNein
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Pythagorean;2406426 But to me said:
But it is....

Physics is a mathematical model of the real world. You take an equation and put limitations on it that are found through observation.

I don't see why people think physics is something more grand or special.
 
  • #181
Pythagorean
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But it is....

Physics is a mathematical model of the real world. You take an equation and put limitations on it that are found through observation.

I don't see why people think physics is something more grand or special.
I'm not making the argument that it's grand or special. But it's more than "mere modeling". It's effective and functional modeling! It's not completely arbitrary and incorrect. It works, and it works well. It helps make predictions about reality. Of course, this couldn't be done accurately without the language of mathematics.
 
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  • #182
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But it is....

Physics is a mathematical model of the real world. You take an equation and put limitations on it that are found through observation.

I don't see why people think physics is something more grand or special.

Agreed.

I know all the arguments for the independent existence of abstract objects, and i still find weird. It is so weird.
 
  • #183
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I'm not making the argument that it's grand or special. But it's more than "mere modeling". It's effective and functional modeling! It's not completely arbitrary and incorrect. It works, and it works well. It helps make predictions about reality. Of course, this couldn't be done accurately without the language of mathematics.
Very vague. "Funcational modeling" don` t give me any insight.

The role of math is in the formulation of physical laws, and the role it plays to carry out the implications of those laws. I find laws of nature to be more useful than the language used to describe it.
 
  • #184
Pythagorean
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Very vague. "Funcational modeling" don` t give me any insight.

The role of math is in the formulation of physical laws, and the role it plays to carry out the implications of those laws. I find laws of nature to be more useful than the language used to describe it.
Functional modeling: Newton's First Law is a case of functional modeling that doesn't require mathematics to comprehend.

Geometry itself was originally very empirical.

Then a perfect model was made in mathematics: The equation of a circle or a square, for instance. These objects don't exist in reality, but they make excellent approximations to our empirical models that they were indeed derived from. We recognize a class of objects that is 'squarish', or 'circlish'.

So the line between mathematics and physics is fuzzy at times. An excellent example is basic addition/subtraction. If there are four enemies in front of us, and we kill one, there is now three. That's a very observationally based model, and it often makes a lot of sense for us to view things as concrete wholes. Integers. This is so ingrained into mathematics that some would think it existed as a theory before it was observed, but I would think that it was observed first, but was so simple to incorporate: "hey! let's give each situation a different name: this will be one, this will be two, and we can add them such that they form the next situation which we'll call three!"

Once a set of rules is put together into a certain classification, the mathematicians will then formulate the rules and push them to the limits, and find errors in the logic and iron them out and make them more consistent. By pushing them to the limits, they will find consequences of implications of the original logic that may lead to or confirm other logic... If it doesn't, we reformulate the rules. In many cases, the full mathematical range of a set of rules is not observable (so we make new mathematical language: "this only works on x>0", but also in many cases, the mathematics makes predictions (based on the original logic, which most likely came from observed phenomena).

We can make up any function we chose in mathematics. It doesn't have to represent any physical observations to be mathematics.
 
  • #185
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Functional modeling: Newton's First Law is a case of functional modeling that doesn't require mathematics to comprehend.

Would the postulates of quantum mechanics be functional modeling?

Geometry itself was originally very empirical.

Then a perfect model was made in mathematics: The equation of a circle or a square, for instance. These objects don't exist in reality, but they make excellent approximations to our empirical models that they were indeed derived from. We recognize a class of objects that is 'squarish', or 'circlish'.

So the line between mathematics and physics is fuzzy at times. An excellent example is basic addition/subtraction. If there are four enemies in front of us, and we kill one, there is now three. That's a very observationally based model, and it often makes a lot of sense for us to view things as concrete wholes. Integers. This is so ingrained into mathematics that some would think it existed as a theory before it was observed, but I would think that it was observed first, but was so simple to incorporate: "hey! let's give each situation a different name: this will be one, this will be two, and we can add them such that they form the next situation which we'll call three!"
fine.

Once a set of rules is put together into a certain classification, the mathematicians will then formulate the rules and push them to the limits, and find errors in the logic and iron them out and make them more consistent. By pushing them to the limits, they will find consequences of implications of the original logic that may lead to or confirm other logic... If it doesn't, we reformulate the rules. In many cases, the full mathematical range of a set of rules is not observable (so we make new mathematical language: "this only works on x>0", but also in many cases, the mathematics makes predictions (based on the original logic, which most likely came from observed phenomena).

It seems to me that you do think there is an initial assignment between the mathematical symbols( 1, 2 ,3 ..), and physical world. You were talking about numbers being assingned to different people. When one person die, we use subtraction, right?

Even in this basic assignment of numbers with people, we are created a semantic map between the math symbols and the real world( people). Notice that we can say:

1) the initial assignment formulated in terms of mathematical symbols. Each person corresponds to a number, etc.

2) The consequence of the mathematical manipulation( ex 2+3=5) makes predictions about the real world( there are 5 people).

This is exactly what i am saying. math give us a precise formulation of laws( in 1), and help us tease out the consequence of those laws( in 2).
 
  • #186
Pythagorean
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It seems to me that you do think there is an initial assignment between the mathematical symbols( 1, 2 ,3 ..), and physical world. You were talking about numbers being assingned to different people. When one person die, we use subtraction, right?

Even in this basic assignment of numbers with people, we are created a semantic map between the math symbols and the real world( people). Notice that we can say:

1) the initial assignment formulated in terms of mathematical symbols. Each person corresponds to a number, etc.

2) The consequence of the mathematical manipulation( ex 2+3=5) makes predictions about the real world( there are 5 people).

This is exactly what i am saying. math give us a precise formulation of laws( in 1), and help us tease out the consequence of those laws( in 2).
Yeah, I don't think we're in disagreement here at all. I was just elaborating in my last post. But as for 2), my point was more focused on the fact that we developed the rules for (a+b=x) based on observation. The mathematics is the language we used to further define the operations we were doing (the + and the =) as well as the numbering system.

My point is that the mathematics transcends reality. The number system goes to negative numbers in mathematics. -4 enemies doesn't make sense in the context of our model. So mathematics makes unreasonable predictions if we don't constrain it based on more observations. In this way, we invent math as we go along to fit our observations.

But math isn't one thing. I'll bet everybody has their own idea of what mathematics is. To me, it's just an academic branch. The "deeper thing" going on in both mathematics and physics is logic.
 
  • #187
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0
Yeah, I don't think we're in disagreement here at all. I was just elaborating in my last post. But as for 2), my point was more focused on the fact that we developed the rules for (a+b=x) based on observation. The mathematics is the language we used to further define the operations we were doing (the + and the =) as well as the numbering system.
I will not make any comment about why 1+2=3, because it will go outside the topic. I will say the assignment in coming up with a math model is the assignment between math symbols with physical quantities.

Ex:

M stands for mass
C stands for light
E stands for energy

so that the model MC^2=E tell us something about the world.


My point is that the mathematics transcends reality. The number system goes to negative numbers in mathematics. -4 enemies doesn't make sense in the context of our model. So mathematics makes unreasonable predictions if we don't constrain it based on more observations. In this way, we invent math as we go along to fit our observations.

I am not going to taking about math transcending physical reality because we can count to -4. Outside the topic.

I would say we use the math to describe our observations. We come to a model by the assigment of math symbols to physical quantities, and we find relationships between the math symbols from generalization in empirical experiments.


The "deeper thing" going on in both mathematics and physics is logic.

I would not say physics is logic. Physics is about trying to know how the world works.
Logic is the study of formal arguments such that true premises lead to true conclusion. They are apples and oranges.
 
  • #188
SixNein
Gold Member
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Functional modeling: Newton's First Law is a case of functional modeling that doesn't require mathematics to comprehend.

Geometry itself was originally very empirical.

Then a perfect model was made in mathematics: The equation of a circle or a square, for instance. These objects don't exist in reality, but they make excellent approximations to our empirical models that they were indeed derived from. We recognize a class of objects that is 'squarish', or 'circlish'.

So the line between mathematics and physics is fuzzy at times. An excellent example is basic addition/subtraction. If there are four enemies in front of us, and we kill one, there is now three. That's a very observationally based model, and it often makes a lot of sense for us to view things as concrete wholes. Integers. This is so ingrained into mathematics that some would think it existed as a theory before it was observed, but I would think that it was observed first, but was so simple to incorporate: "hey! let's give each situation a different name: this will be one, this will be two, and we can add them such that they form the next situation which we'll call three!"

Once a set of rules is put together into a certain classification, the mathematicians will then formulate the rules and push them to the limits, and find errors in the logic and iron them out and make them more consistent. By pushing them to the limits, they will find consequences of implications of the original logic that may lead to or confirm other logic... If it doesn't, we reformulate the rules. In many cases, the full mathematical range of a set of rules is not observable (so we make new mathematical language: "this only works on x>0", but also in many cases, the mathematics makes predictions (based on the original logic, which most likely came from observed phenomena).

We can make up any function we chose in mathematics. It doesn't have to represent any physical observations to be mathematics.
Integers is a modern concept. Integers feel natural now, but they were hard to get accepted at first. There is no real observation of negative distance, negative apples, or anything of the kind; however, integers really revolutionized money and more importantly debt. Even the number zero took time. Why would a farmer need to count zero sheep? After he or she lost all sheep, he or she was out of business anyway.

Mathematics still works by and large from observations. Many mathematicians reword problems into another form that can be observed or visualized. A good example would be p=np. If you could solve the traveling salesman problem, then it would automatically answer the p=np question.
 
  • #189
Pythagorean
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Integers is a modern concept. Integers feel natural now, but they were hard to get accepted at first. There is no real observation of negative distance, negative apples, or anything of the kind; however, integers really revolutionized money and more importantly debt. Even the number zero took time. Why would a farmer need to count zero sheep? After he or she lost all sheep, he or she was out of business anyway.
I think this is a mathematical perspective. From an observational perspective, integers were an easy concept. Babies and monkeys alike can tell when something they want is missing, they can tell the difference between 2 and 3 cookies. It's very natural, I think, for us to take account of these things, starting with the emotional framework, "I want more of what I like, rather than less of what I like". From this more/less concept arises counting in our later development.

Studying integers as mathematical objects, I agree, is a more modern concept.

A speculative citation: The first known use of numbers, 30,000 BC, was tally marks, an integer number system:
http://en.wikipedia.org/wiki/Number#History_of_integers
 
  • #190
SixNein
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I think this is a mathematical perspective. From an observational perspective, integers were an easy concept. Babies and monkeys alike can tell when something they want is missing, they can tell the difference between 2 and 3 cookies. It's very natural, I think, for us to take account of these things, starting with the emotional framework, "I want more of what I like, rather than less of what I like". From this more/less concept arises counting in our later development.

Studying integers as mathematical objects, I agree, is a more modern concept.

A speculative citation: The first known use of numbers, 30,000 BC, was tally marks, an integer number system:
http://en.wikipedia.org/wiki/Number#History_of_integers
I would agree that natural or whole numbers can be observed, but integers are a different story. Can a monkey tell that he's got negative apples? No apples and negative apples are different concepts.

I'm inclined to believe that natural numbers are older than mankind. I would even venture a wild theory that dinosaurs could do some kind of counting. The concept of integers, however, separates us from the wild.
 
  • #191
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Aristotle's Law of Inertia said that a body comes to its natural state of rest unless acted upon by an outside force. By the Renaissance no counter example had ever been observed. Yet Gallileo said that a body in motion will stay in motion unless acted upon by an outside force. His evidence was the inclinded plane experiments. But as he was dersively told by his comtemporaries, the balls eventually come to a state of rest and in fact do not rise to the same height. So there was another ingredient other than just modeling observation in Gallileo's thinking and in fact in his day his theory contradicted observation while Aristotle's did not.
 
  • #192
Pythagorean
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I would agree that natural or whole numbers can be observed, but integers are a different story. Can a monkey tell that he's got negative apples? No apples and negative apples are different concepts.

I'm inclined to believe that natural numbers are older than mankind. I would even venture a wild theory that dinosaurs could do some kind of counting. The concept of integers, however, separates us from the wild.
Oh, of course I've been using the wrong word! I meant natural/whole numbers from the beginning. I've always had a tendency to call natural numbers integers because we use the word commonly in physics for n = 1, 2, 3, etc. I now see that you mean to exclude 0 and negative numbers (and probably infinity too?) which I would agree with. Of course, I'm sure zero was observed early on, but it wasn't something that was discussed or recognized, as you may have implied earlier.
 
  • #193
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Mathematics still works by and large from observations.
Can you show me how for vector spaces, and group theory?

What about algebraical notions like fields, rings etc?
 
  • #194
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His evidence was the inclinded plane experiments. But as he was dersively told by his comtemporaries, the balls eventually come to a state of rest and in fact do not rise to the same height. So there was another ingredient other than just modeling observation in Gallileo's thinking and in fact in his day his theory contradicted observation while Aristotle's did not.
How is that? His`s only claim is that acceration ( or g) is the same for all large, and smell things. I don ` t see why this is NOT a observation generalized law.
 
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