Why does math work in our reality?

[27Thousand]

These are all scientific theories... Math is a different ball park. Yes, science uses math... no math is not science.

Wait, so I'm confused here If math can "describe" observable principles, and make "predictions" for observations, how does it not work in our reality? It may not explain the way a scientific theory does, and may not actually be an observable principle itself (scientific law), but if it does a very good job of describing and predicting the observable, how is it not tied into some sort of reality? When I take a statistics class, math can accurately predict a range for probability in a bell curve, or describe what's already happened.

 

[Chronos]

Math is pretty scary. It took us in directions we never imagined throughout human history, and has never been refuted. I view it as a very rigorous form of logic built from the ground up. It implies we live in a logical universe. That remains to be seen. but. still looks like a good bet. If the universe truly is illogical, I doubt we will ever comprehend it.

 

[fatra2]

Hi there,

Wait, so I'm confused here If math can "describe" observable principles, and make "predictions" for observations, how does it not work in our reality?

I often give the following example to explain the usefulness of mathematics.

If you want to build a house, you will need a hammer (out of many other tools). The reasons a hammer works is very simple: a hammer is design and made to nail stuff in a wall.

You should mathematics in the same way. Mathematics are tools to help us model some theory that tries to explain something real. This tool is design and made to model these theories. Therefore, mathematics evolved over the years. When the known mathematics are not enough to explain this or that, scientist or mathematician will develop some more theories.

Cheers

 

[wofsy]

There is a reverse way of looking at it. Our observations are not reality but merely reality as presented through our senses. Through science we start with our sense impressions and discover what reality is really like underneath. Mathematics is one tool for doing this.

Why can't we view science as a way of perceiving reality that is otherwise only partially known through the senses? Why are the senses so sacrosanct?

 

[M Grandin]

For instance "HallsofIvy" and "Junglebeast" have already answered the OP question fully satisfactory, why I have not much to add. But expressed in own words, I could say
mathematics is just systematisized logics, where logics operating on logics may result in impressing formulas and mathematical complex using symbols resembling alien language to
common people. All resting on elementary building stones of logics and fundamental observations = axioms. These axioms may look like abstractions without regard to real world
- but may be more of physical observation than abstract thinking than people believe.

So because these fundamental axioms and logics in fact are fundamental experienced "physics", using these systematically may also result in something matching real world.

 

[wofsy]

For instance "HallsofIvy" and "Junglebeast" have already answered the OP question fully satisfactory, why I have not much to add. But expressed in own words, I could say
mathematics is just systematisized logics, where logics operating on logics may result in impressing formulas and mathematical complex using symbols resembling alien language to
common people. All resting on elementary building stones of logics and fundamental observations = axioms. These axioms may look like abstractions without regard to real world
- but may be more of physical observation than abstract thinking than people believe.

So because these fundamental axioms and logics in fact are fundamental experienced "physics", using these systematically may also result in something matching real world.

that is interesting - I experience mathematics not only as axioms and deductions but also as a branch of science. Mathematicians certainly do not think of themselves as mere logicians.

I think that there are mathematical objects of empirical study just as there are physical ones. There are mathematical theories just as there are biological or physical. Mathematical ideas require incredible imagination and are often derived from observation of mathematical objects and relationships - just as in any science.

The mathematics that is used for instance in General Relativity was first invented by mathematicians who were challenging our ideas of measurement and of intrinsic geometry. They came up with new theories which later - happened to have application in physics. A modern example is Chern-Simons invariants which were discovered during pure geometrical researches and later were found to have application in particle physics.

 

[kote]

that is interesting - I experience mathematics not only as axioms and deductions but also as a branch of science. Mathematicians certainly do not think of themselves as mere logicians.

I think that there are mathematical objects of empirical study just as there are physical ones. There are mathematical theories just as there are biological or physical. Mathematical ideas require incredible imagination and are often derived from observation of mathematical objects and relationships - just as in any science.

The mathematics that is used for instance in General Relativity was first invented by mathematicians who were challenging our ideas of measurement and of intrinsic geometry. They came up with new theories which later - happened to have application in physics. A modern example is Chern-Simons invariants which were discovered during pure geometrical researches and later were found to have application in particle physics.

I'm not sure what the "mere" is for when you refer to logicians. If mathematics is the science of discovering real mathematical objects, the implication is that mathematical theorems, like in science, can be wrong, and proofs are not actually proofs but just hypotheses. Is this your stance?

If math is a science there is no such thing as mathematical proof and we should rewrite all of the textbooks. We should also allow for inconsistent mathematical theories, as is done in science, and not automatically accept "proofs" against the consistency of theories.

The application of math to the world is science, so the fact that math is used in scientific theories is irrelevant to the math itself. Euclidean space is just as valid as Minkowski space. Whether or not one gives a better model of reality is outside the realm of mathematics.

Math is either deduction from axioms or an inductive science. It can't be both.

 

[wofsy]

I'm not sure what the "mere" is for when you refer to logicians. If mathematics is the science of discovering real mathematical objects, the implication is that mathematical theorems, like in science, can be wrong, and proofs are not actually proofs but just hypotheses. Is this your stance?

If math is a science there is no such thing as mathematical proof and we should rewrite all of the textbooks. We should also allow for inconsistent mathematical theories, as is done in science, and not automatically accept "proofs" against the consistency of theories.

The application math to the world is science, so the fact that math is used in scientific theories is irrelevant to the math itself. Euclidean space is just as valid as Minkowski space. Whether or not one gives a better model of reality is outside the realm of mathematics.

Math is either deduction from axioms or an inductive science. It can't be both.

My point is that mathematics is empirical and studies empirical objects just as any other science. Mathematicians even do experiments. They have the further more powerful tool that their theories can be substantiated by proof. If having this tool means that these other mathematical activities - including incredibly creative ideas - makes it not a science - then I think that is a definition - one that Gauss for instance did not agree with.

Math is both inductive and deductive. take this scenario. A person wants to know whether a certain geometrical property holds for a class of Riemannian manifolds. There are infinitely many such manifolds and they exist in all finite dimensions. Few examples are known and all of them are in low dimensions. So what does this person do? Does he try to deduce the answer? Maybe. But more likely he will start to look at examples. Based on these examples he will form hypotheses that he will check in other examples. If these hypotheses fail he will either modify them or look for new relationships and come up with new hypotheses and check them out again. Eventually he will find a property, a mathematical relationship, that reveals the truth or falseness of his original question. He then may consider the relationship that he has found ,though giving the answer, may not satisfactorily reveal how the property in question relates to broader questions of ongoing research. Thus he may revisit his investigation in search of other properties that allow this broader understanding. To me, this is science. the thought processes are the same.

the attitude that I have found is that mathematicians and physicists view the two as branches of the same subject. One PDE researcher said to me that his mathematical research though pure and not pointed at any scientific endeavor nevertheless examines certain geometrical minimization problems which he believes to relate to intrinsic features of the universe.

 

[kote]

My point is that mathematics is empirical and studies empirical objects just as any other science. Mathematicians even do experiments. They have the further more powerful tool that their theories can be substantiated by proof. If having this tool means that these other mathematical activities - including incredibly creative ideas - makes it not a science - then I think that is a definition - one that Gauss for instance did not agree with.

Science is the inductive exploration of reality. Sciences don't have proofs, they have experiments and inductive generalizations. Mathematicians can do science. They can perform experiments. They can be creative. But the meta-activity of mathematicians is as much math as Einstein's morning teeth-brushing routine is science.

If math is mere science then proofs of 2+2=4 are invalid. That has been a popular claim on here though.

 

[wofsy]

Science is the inductive exploration of reality. Sciences don't have proofs, they have experiments and inductive generalizations. Mathematicians can do science. They can perform experiments. They can be creative. But the meta-activity of mathematicians is as much math as Einstein's morning teeth-brushing routine is science.

If math is mere science then proofs of 2+2=4 are invalid. That has been a popular claim on here though.

I expanded my note to you. what is your reaction?

 

[wofsy]

Science is the inductive exploration of reality. Sciences don't have proofs, they have experiments and inductive generalizations. Mathematicians can do science. They can perform experiments. They can be creative. But the meta-activity of mathematicians is as much math as Einstein's morning teeth-brushing routine is science.

If math is mere science then proofs of 2+2=4 are invalid. That has been a popular claim on here though.

Einstein did not brush his teeth.

 

[wofsy]

Science is the inductive exploration of reality. Sciences don't have proofs, they have experiments and inductive generalizations. Mathematicians can do science. They can perform experiments. They can be creative. But the meta-activity of mathematicians is as much math as Einstein's morning teeth-brushing routine is science.

If math is mere science then proofs of 2+2=4 are invalid. That has been a popular claim on here though.

A subject that uses proof can still also use science. I do not believe that they are mutually exclusive.

 

[kote]

I expanded my note to you. what is your reaction?

Sure, sociologically there is a method involved in how mathematicians decide what they want to prove. Whether a mathematician has a grant to investigate something and form a conjecture or whether the grant is to write a proof is not how I think we should draw the line. This is similar to how creatively coming up with new theories in science, theories for which no experiments have yet been done, is not part of anyone's definition of the scientific method. The science is in the experiments, not the conjecture, and not the sociological or psychological motivations.

I prefer to draw the line at inductive/deductive and idealized notions rather than looking at all of the sociological factors involved. This avoids problems such as the question of whether or not writing grant proposals is "math" or "science." It also avoids the fact that mathematicians are not infallible. How do we know that a proof is a proof and we haven't made a mistake? Well... we don't, but that's a fault of mathematicians and not of math.

 

[wofsy]

Sure, sociologically there is a method involved in how mathematicians decide what they want to prove. Whether a mathematician has a grant to investigate something and form a conjecture or whether the grant is to write a proof is not how I think we should draw the line. This is similar to how creatively coming up with new theories in science, theories for which no experiments have yet been done, is not part of anyone's definition of the scientific method. The science is in the experiments, not the conjecture, and not the sociological or psychological motivations.

I prefer to draw the line at inductive/deductive and idealized notions rather than looking at all of the sociological factors involved. This avoids problems such as the question of whether or not writing grant proposals is "math" or "science." It also avoids the fact that mathematicians are not infallible. How do we know that a proof is a proof and we haven't made a mistake? Well... we don't, but that's a fault of mathematicians and not of math.

I do not think that the process of mathematical thought and investigation is sociological. It is a necessary aspect of mathematical ideation. the thought processes are fundamentally scientific. The experimenter does an experiment to verify a hypothesis or to examine a property of a physical system. A mathematician examines mathematical objects for the same reason, to verify a hypothesis or to examine a property. No difference.

While it is true that no proof can ever be know for sure - neither can the result of any experiment be know to be always repeatable. If experimenters did not believe that their evidence represents something immutable and invariant - they would never have a theory of anything. It is true that in science this is a belief - an act of faith perhaps- whereas in mathematics it is not.

 

[apeiron]

Math is either deduction from axioms or an inductive science. It can't be both.

And where do axioms come from if not by induction? General ideas derived from particular impressions.

It is also obvious (since Godel at least) that all axioms demand an epistemic cut - the arbitrary insertion of an observer. At some point it is decided that all this is true, because all that is false. A crisp choice gets made. So even as generalities, axioms are always going to be subsets of the possible. A choice is made and stuff must get left behind. Or better yet, as good epistemic cuts are formally dichotomous, middles get excluded.

Have you read Robert Rosen or Howard Pattee? They have written good stuff on these matters.

 

[kote]

And where do axioms come from if not by induction? General ideas derived from particular impressions.

It is also obvious (since Godel at least) that all axioms demand an epistemic cut - the arbitrary insertion of an observer. At some point it is decided that all this is true, because all that is false. A crisp choice gets made. So even as generalities, axioms are always going to be subsets of the possible. A choice is made and stuff must get left behind. Or better yet, as good epistemic cuts are formally dichotomous, middles get excluded.

Have you read Robert Rosen or Howard Pattee? They have written good stuff on these matters.

Well of course we choose axioms to give us systems that are useful in science etc, but the part about choosing axioms isn't math ! I agree with what you said, I just see a choice of axioms as meta-math. Axioms are chosen on decidedly non-mathematical grounds like aesthetics or a presupposed scientific utility. Justification of consistent axioms is not itself mathematical.

I'll look into those authors.

 

[apeiron]

Ok, so axioms go back in the philosophy bin!

All meta- level discussions are philosophical because that is the place for vague deliberations (as opposed to the crisply taken choices of maths and science).

We are talking about how we know the world. Philosophy is where the vague groping exploration of possibilities take place. Then when it comes to "method", maths is for formalising the modelling and science is for formalising the measurements.

Actually "doing science" of course involves all three. We are in a modelling relation with reality (see Rosen). We start out with vague ideas and impressions and attempt to develop them into a crisp system of models and measurements.

Philosophy gets us started. Then we start to take the choices that swim into view.

Maths is a way of creating definite models. Self-consistent frameworks that clearly, crisply, represent our ideas about causality.

Scientific method then sets out the looping process of measurement, the optimal system for generalising the necessary acts of observation. The apparatus of experiment and hypothesis, etc.

Maths works not because of some platonic magic but because reality is itself a collection of interactions that must settle into emergent patterns. There is a reduction of possibility that takes place "out there". And we are trying to do the same thing in our own minds.

I would argue that so far we have only really been doing half the job with the maths we've produced though.

We have a very well developed mathematics of atomism, a very poor mathematics of systems.

If you study hierarchy theory and other tentative examples of systems maths, they are indeed more "philosophical" - vaguely developed ideas rather than crisply taken choices.

But with chaos theory, tsallis entropy, fractal geometry, renormalisation group and scalefree networks, for example, systems math is starting to emerge in earnest.

The key is the reintroduction of scale into mathematics. Atomistic maths excluded scale. A platonic triangle is a scale-less concept. It could be any size. But a Sierpinski gasket is a triangle with scale. We now have an axis of scale symmetry being represented, a open system or powerlaw realm.

So mathematics works because it is making crisp what was vaguely seen in philosophy. It works because reality is self-organising pattern. It works because it split off the model making issues from the measurement taking issus.

But it's job is far from complete. Atomism is well elaborated. But the field of systems mathematics is just in the process of being born.

 

[Hippasos]

One amazing example observed physical behavior of quantum chaos conjectured by the pure mathematics area of number theory:

http://www.physorg.com/news142834558.html

 

[wofsy]

Ok, so axioms go back in the philosophy bin!

All meta- level discussions are philosophical because that is the place for vague deliberations (as opposed to the crisply taken choices of maths and science).

We are talking about how we know the world. Philosophy is where the vague groping exploration of possibilities take place. Then when it comes to "method", maths is for formalising the modelling and science is for formalising the measurements.

Actually "doing science" of course involves all three. We are in a modelling relation with reality (see Rosen). We start out with vague ideas and impressions and attempt to develop them into a crisp system of models and measurements.

Philosophy gets us started. Then we start to take the choices that swim into view.

Maths is a way of creating definite models. Self-consistent frameworks that clearly, crisply, represent our ideas about causality.

Scientific method then sets out the looping process of measurement, the optimal system for generalising the necessary acts of observation. The apparatus of experiment and hypothesis, etc.

Maths works not because of some platonic magic but because reality is itself a collection of interactions that must settle into emergent patterns. There is a reduction of possibility that takes place "out there". And we are trying to do the same thing in our own minds.

I would argue that so far we have only really been doing half the job with the maths we've produced though.

We have a very well developed mathematics of atomism, a very poor mathematics of systems.

If you study hierarchy theory and other tentative examples of systems maths, they are indeed more "philosophical" - vaguely developed ideas rather than crisply taken choices.

But with chaos theory, tsallis entropy, fractal geometry, renormalisation group and scalefree networks, for example, systems math is starting to emerge in earnest.

The key is the reintroduction of scale into mathematics. Atomistic maths excluded scale. A platonic triangle is a scale-less concept. It could be any size. But a Sierpinski gasket is a triangle with scale. We now have an axis of scale symmetry being represented, a open system or powerlaw realm.

So mathematics works because it is making crisp what was vaguely seen in philosophy. It works because reality is self-organising pattern. It works because it split off the model making issues from the measurement taking issus.

But it's job is far from complete. Atomism is well elaborated. But the field of systems mathematics is just in the process of being born.

How do you you know a prioi what reality is or isn't?

How do you know that reality is a "self collection of interactions" (whatever that means). Whatever that is supposed to measan, isn't that a model - not a very crisp one though - maybe a meaningless one.

Mathematics does not work because it is making crisp (what ever that means) what is vague in philosophy. You do not know why mathematics works. Nobody does.

"The key is the reintroduction of scale into mathematics. Atomistic maths excluded scale." This is meaningless - and how do you know what the key is?

"Scientific method then sets out the looping process of measurement, the optimal system for generalising the necessary acts of observation." What is this supposed to mean?

Science is not a system - it does not generalize observation - "generalizing observation" is an oxymoron.

Chaos theory and fractal math have little influence on scientific research. Both are fads. How then do you know then that these are the right direction of science?

"The key is the reintroduction of scale into mathematics. Atomistic maths excluded scale. A platonic triangle is a scale-less concept. It could be any size. But a Sierpinski gasket is a triangle with scale. We now have an axis of scale symmetry being represented, a open system or power law realm." Ahat is this supposed to mean? Any scientist who heard you say this would smile politely then walk away. Why don't you go to a physics department and try it out on a mathematical physicist?

"Philosophy is where the vague groping exploration of possibilities take place." Not true. What is vague groping? Can you make that more crisp?

"Then when it comes to "method", maths is for formalising the modelling and science is for formalising the measurements." That is wrong. Math is not for formalizing nor is science. Formalizing always occurs after the science and math have already been done. Axioms have little to do with scientific thinking. They are afterthoughts.

'Maths is a way of creating definite models. Self-consistent frameworks that clearly, crisply, represent our ideas about causality. " Causality is not what math studies.In fact ideas of causality have always been an impediment to physics and science has always tried to eliminate causality in order to make progress.
"

Math is not a way of creating definite models - models are formal devices - real mathematics is a way of discovering our ideas of space, geometry and number- empirical observation can guide these discoveries but it is not the only source of guidance. What empirical model of reality would you say the theory of Riemann surfaces represents? How about the theory of differentiable structures on manifolds? Which empirical data did the Riemann hypothesis model? - what observations did it make "crisp"?
How about Thom's theory of cobordism of differentiable manifolds? After you explain all of these to me, you can move on to Chern-Simons invariants and then rational homotopy theory/ Oh yeah and maybe you could help me out with which empirical data the theory of Bieberbach groups was designed to model.

Even the general theory of relativity did model model anything new - it was a reconceptualization of our ideas of space. Only after it was discovered was it found to predict certain new data that previous theories did not.

Science develops because people question or ideas of reality not because we model it. The Ptolememaic system was a great model of planetary motion. Yet it was questioned - not for empirical reasons but because people felt that it could not be consistent with the mind of God. When Gallileo said that the ball rolling on an inclined plane would rise to the same height he was discovering an idea of reality not explaining empirical data. In fact, people said to him that he was wrong because the ball did not rise exactly to the same height and the more it rolled back and forth the less it rose until it finally came to a stop. People said that on the contrary this confirmed Aristotle's patently accurate model of reality which was that an object in motion will come to its natural state of rest. the empirically correct model contradicted Gallileo's conclusion. His model was empirically false. Yet he said, 'If God wanted me to be wrong he would have made the ball miss by a mile not by an inch.'

 

[kote]

Science is not a system - it does not generalize observation - "generalizing observation" is an oxymoron.

I'm confused here. Generalizing observation is exactly what science does. You take measurements and generate models or theories. Then, you check to make sure your theory fits all of the generalizations you intended it to fit by making additional observations.

Math is not for formalizing nor is science. Formalizing always occurs after the science and math have already been done. Axioms have little to do with scientific thinking. They are afterthoughts.

I suppose you'll have to tell that to Einstein regarding an assumption of invariance or QM regarding the Heisenberg Uncertainty Principle. Relativity theory is the formalization resulting from the axiom that the laws of physics are the same in any intertial reference frame.

In fact ideas of causality have always been an impediment to physics and science has always tried to eliminate causality in order to make progress.

Really? Science doesn't try to uncover the patterns between causes and effects? It doesn't try to tell you why certain things are observed? Are you saying that the less science has to do with physical cause and effect, the better it is? Physics is the study of physical causation.

In fact, people said to him that he was wrong because the ball did not rise exactly to the same height and the more it rolled back and forth the less it rose until it finally came to a stop. People said that on the contrary this confirmed Aristotle's patently accurate model of reality which was that an object in motion will come to its natural state of rest. the empirically correct model contradicted Gallileo's conclusion. His model was empirically false. Yet he said, 'If God wanted me to be wrong he would have made the ball miss by a mile not by an inch.'

Aristotle's model didn't make predictions. Gallileo's model was empirically more accurate. Being within an inch is closer than not even making a guess.

You posted a lot, so I'll have to apologize for only responding to parts.

 

[wofsy]

I'm confused here. Generalizing observation is exactly what science does. You take measurements and generate models or theories. Then, you check to make sure your theory fits all of the generalizations you intended it to fit by making additional observations.
I suppose you'll have to tell that to Einstein regarding an assumption of invariance or QM regarding the Heisenberg Uncertainty Principle. Relativity theory is the formalization resulting from the axiom that the laws of physics are the same in any intertial reference frame.
Really? Science doesn't try to uncover the patterns between causes and effects? It doesn't try to tell you why certain things are observed? Are you saying that the less science has to do with physical cause and effect, the better it is? Physics is the study of physical causation.
Aristotle's model didn't make predictions. Gallileo's model was empirically more accurate. Being within an inch is closer than not even making a guess.

You posted a lot, so I'll have to apologize for only responding to parts.

Ok Aristotle's model worked better
-science does not generalize observation-- observation is merely sense data.
Science does not fundamentally uncover patterns - pattern fitting is only a tool -what about the Ptolemaic system? - a great pattern fit. It was considered to be dogmatic, axiomatic, rigid and false because it did not penetrate our ideas of how God the geometer would have constructed the universe.

- Einstein's theory of relativity originally did not,at first, explain new data - it discovered a new concept of space time - physicists believe that is was a totally unexpected and anomalous idea precisely because of this. They agree that physics could have gone along explaining known data just fine without it.

Most mathematical theories examine our ideas of space and quantity. I gave only a few examples. I think it would benefit the philosopher to do some real mathematics.

I guess underlying these thoughts is the idea that sense perception is merely a shadow of reality and its true nature needs to be discovered with this shadow as a guide. These is no doubt that science progressed on this assumption. The physicists and mathematicians who believed this are too numerous to list - e.g. Gauss, Riemann, Einstein, Kepler, Planck,most Quantum physicists, etc

There is a deep book on this subject written by Henri Poincare called Science and Hypothesis. In it he says that there are two schools of physics, the English school which denies the need for hypotheses about the nature of reality and believes that all science is just pattern fitting and the Continental school which says that underlying assumptions/hypotheses are necessary. the pattern fitters were notably Newton and Faraday both who explicitly claimed that they needed no hypotheses - maybe Maxwell also though I am not sure of his view on this. Curiously it seems that there were really two schools of physics centered around this controversy.

 

[kote]

Science does not fundamentally uncover patterns - pattern fitting is only a tool -what about the Ptolemaic system? - a great pattern fit. It was considered to be dogmatic, axiomatic, rigid and false because it did not penetrate our ideas of how God the geometer would have constructed the universe.

What is science if not a tool? Also, the Ptolemaic system is not "false," it is simply inelegant. Inelegance is not a mathematical or scientific criteria; it is aesthetics. And since when is a historical religious view the determining factor in scientific worth? Is acceptance by the church a requirement of good science?

Maybe it would help if you could explain what you think the goal of science is if it is not to formally generalize observations or to uncover physical causation.

I think it would benefit the philosopher to do some real mathematics.

Why do you assume we haven't done "real math?" What does performing calculations or derivations or constructing proofs have to do with the philosophical foundations of mathematics anyway?

 

[wofsy]

What is science if not a tool? Also, the Ptolemaic system is not "false," it is simply inelegant. Inelegance is not a mathematical or scientific criteria, it is aesthetics. And since when is a historical religious view the determining factor in scientific worth? Is acceptance by the church a requirement of good science?

Maybe it would help if you could explain what you think the goal of science is if it is not to formally generalize observations or to uncover physical causation.
Why do you assume we haven't done "real math?" What does performing calculations or derivations or constructing proofs have to do with the philosophical foundations of mathematics anyway?

By doing mathematics - by which I mean coming up with new theories or extending existing theories in important ways - not just doing proofs and calculations - you will see my point.
If you have done this and don't agree with me then I am astounded and would like to discuss it further. Do you think the theory of Riemann surfaces was thought of as a calculation or a proof of something?

Science is a tool but is more than that. that is what I am trying to say. I am also saying that the idea of mathematics as merely a modelling tool is just plain wrong. If you are a mathematician, how is it that you believe this?

By the way, the Ptolemaic system - though a good pattern fit, a grand lovely model - was an impediment to the progress of science - yet it was able to predict and to model wonderfully.

 

[kote]

By doing mathematics - by which I mean coming up with new theories or extending existing theories in important ways - not just doing proofs and calculations - you will see my point.
If you have done this and don't agree with me then I am astounded and would like to discuss it further. Do you think the theory of Riemann surfaces was thought of as a calculation or a proof of something?

Science is a tool but is more than that. that is what I am trying to say. I am also saying that the idea of mathematics as merely a modelling tool is just plain wrong. If you are a mathematician, how is it that you believe this?

It's because the idea that modeling deserves to be qualified by "mere" is a philosophical idea and not a mathematical one. Judgments about the value of modeling are not mathematical or scientific. I disagree that math is a science, so I see no problem in asserting that science is modeling while math is not. Math is a tool used by science to create models. Mathematical theories, in a way, can be said to be models themselves, they just aren't models of anything in particular without science attached. Math is the logical analytic extension of axioms or assumptions.

Mathematical theories often precede scientific theories. They are often surprising and paradoxical and cause us to think about things in new ways. Math is necessary for formal science and nearly all technology. Extending or applying mathematics can be a very creative process. None of that changes the fact that it involves only purely formal structures (or models), the application of which lies in the realm of science, and the value of which lies in the realm of ethics and aesthetics.

From Barry Mazur http://www.math.harvard.edu/~mazur/papers/plato4.pdf:

If we adopt the Platonic view that mathematics is discovered, we are suddenly in surprising territory, for this is a full-fledged theistic position. Not that it necessarily posits a god, but rather that its stance is such that the only way one can adequately express one’s faith in it, the only way one can hope to persuade others of its truth, is by abandoning the arsenal of rationality, and relying on the resources of the prophets. ...

It seems to me that, in the hands of a mathematician who is a determined Platonist, proof could very well serve primarily this kind of rhetorical function—making sure that the description is on track—and not (or at least: not necessarily) have the rigorous theory-building function it is often conceived as fulfilling.​

Only math that involves "mere" abstract analytic constructions can talk about proof or universals. If math discusses anything other than models, it is reduced to mere inductive generalization, which cannot rationally be demonstrated as true.

Mazur also criticizes anti-platonistic views, but he focuses on the idea that math is socially constructed, which is not what we're talking about.

I also object to the view that what a mathematician says he thinks he is doing is valid evidence for what math actually is, unless truth is sociologically constructed. But if truth is sociological, math is already in trouble and needs to be knocked down a few levels.

 

[wofsy]

None of that changes the fact that it involves only purely formal structures (or models),

that to me is just wrong unless I don't know what you mean by purley formal. Are all ideas purely formal? Is love purely formal?

Ideas are not formalisms to me. Is a Bach fugue purely formal if we think of it rather than listen to it. Is the painting in Van Gogh's mind purely formal before he paints it?

 

[wofsy]

Mathematics is discovered. Mazur is right that this has potentially theological implications - but in my mind he is wrong that it is theistic. The reality of ideas is indisputable. Sense experience is always in doubt.

 

[kote]

Mathematics is discovered. Mazur is right that this has potentially theological implications - but in my mind he is wrong that it is theistic. The reality of ideas is indisputable. Sense experience is always in doubt.

Ah, so you are a dualist ! I almost agree with what you're saying here, except for one minor point.

You say that the reality of ideas is indisputable. I agree. But for real ideas to be discovered as opposed to being invented or constructed, they must be real before they are ideas in the minds of any particular mathematician. This leads you to Berkeley's argument that for these ideas to have timeless existence, they must be existing in the mind of God.

If ideas are real, then either they are created in the minds of those who are thinking them, or they exist permanently and objectively in an eternal mind. In order for real ideas to be discovered, there must be an eternal mind in which they exist prior to their being discovered by our minds. Also, in order for ideas to be objective, they must exist in an omniscient objective mind.

If math is ideas that are discovered, then there exists an objective and eternal mind.

If math is relations and not ideas, then we don't have this problem. Analytic relations may be discovered, but they are not objects and do not have such a thing as existence. That is what I mean by the idea that they are purely formal. Their method of discovery is deductive and logical rather than scientific.

 

[wofsy]

Ah, so you are a dualist ! I almost agree with what you're saying here, except for one minor point.

You say that the reality of ideas is indisputable. I agree. But for real ideas to be discovered as opposed to being invented or constructed, they must be real before they are ideas in the minds of any particular mathematician. This leads you to Berkeley's argument that for these ideas to have timeless existence, they must be existing in the mind of God.

If ideas are real, then either they are created in the minds of those who are thinking them, or they exist permanently and objectively in an eternal mind. In order for real ideas to be discovered, there must be an eternal mind in which they exist prior to their being discovered by our minds. Also, in order for ideas to be objective, they must exist in an omniscient objective mind.

If math is ideas that are discovered, then there exists an objective and eternal mind.

If math is relations and not ideas, then we don't have this problem. Analytic relations may be discovered, but they are not objects and do not have such a thing as existence. That is what I mean by the idea that they are purely formal. Their method of discovery is deductive and logical rather than scientific.

good explanation> I understand what you mean. I think our difference is largely semantic. A couple things though and this confuses me a bit. First of all why aren't relations just as objective as ideas?

Second, if we discover them did they come into existence at the moment of discovery or did we only become aware of them? If they come into existence then is the same relation in someone else's thoughts a different relation and if it is what unifies the two? In which mind does this unity exist and was that also created on the spot of discovery?

I like Riemann's idea that the universe itself has the intrinsic dynamics of a mind and that our minds partake in this and are part of it. He was I think a neo-Kantian and believed in intrinsic ideas. But it seems that he tried to extend this to explanation of physical phenomena as thoughts -e.g. particles. from this point of view there is a universal mind in a sense - but not a deity.

 

[kote]

good explanation> I understand what you mean. I think our difference is largely semantic. A couple things though and this confuses me a bit. First of all why aren't relations just as objective as ideas?

It's been argued that all of philosophy is semantics . I'm inclined to agree. I would say that logical relations are objective. By objective here I mean not dependent on any particular point of view. I don't mean that they are objects or have existence.

Ideas are personal to the person who has them. They are about as subjective as you can get. Objectivity is trickier. It can be doubted whether or not there is such a thing at all. When it comes to logical or analytic truths, the best argument I can give for their objectivity (or universality) is that we can't conceive of them as being false or dependent on our point of view. Does it depend on my point of view that something can't both exist and not exist at the same time? Or that no unmarried men have wives?

Logic is the system in which we can make philosophical arguments, and, at least since Godel, we know that no logical system can prove its own validity. The fact that we can't rationally deny that 1=1, though, makes it as objective as may be possible in my opinion. This may also be a misuse of objective and subjective though, in that logic is supposed to come before either of those notions as a framework.

I like Riemann's idea that the universe itself has the intrinsic dynamics of a mind and that our minds partake in this and are part of it. He was I think a neo-Kantian and believed in intrinsic ideas. But it seems that he tried to extend this to explanation of physical phenomena as thoughts -e.g. particles. from this point of view there is a universal mind in a sense - but not a deity.

I'm not familiar with Reimann's philosophy, but it might be relevant to point out that Kant did not believe in intrinsic ideas. From http://plato.stanford.edu/entries/kant-judgment/:

Kantian innateness is essentially a procedure-based innateness, consisting in an a priori active readiness of the mind for implementing rules of synthesis, as opposed to the content-based innateness of Cartesian and Leibnizian innate ideas, according to which an infinitely large supply of complete (e.g., mathematical) beliefs, propositions, or concepts themselves are either occurrently or dispositionally intrinsic to the mind. But as Locke pointed out, this implausibly overloads the human mind's limited storage capacities.​

In other words, logic is innate, ideas are not. I like Bertrand Russell's treatment of the issue. Russell's laws of thought are somewhat similar to Kant's categories of perception. He talks about universals and relations in http://www.ditext.com/russell/russell.html chapters 7-10.

Second, if we discover them did they come into existence at the moment of discovery or did we only become aware of them? If they come into existence then is the same relation in someone else's thoughts a different relation and if it is what unifies the two? In which mind does this unity exist and was that also created on the spot of discovery?

When we discover them, they come into existence as thoughts in our mind. From Russell (chap 9):

It is largely the very peculiar kind of being that belongs to universals which has led many people to suppose that they are really mental. We can think of a universal, and our thinking then exists in a perfectly ordinary sense, like any other mental act. Suppose, for example, that we are thinking of whiteness. Then in one sense it may be said that whiteness is 'in our mind'. We have here the same ambiguity as we noted in discussing Berkeley in Chapter IV. In the strict sense, it is not whiteness that is in our mind, but the act of thinking of whiteness.​

As for unity with other people's thoughts, there is still trouble in overcoming solipsism. There's not even a guarantee that our own thoughts are true representations of logical relations. Proofs have certainly been shown to be wrong before. The best we can probably do is say that unless we accept solipsism, we must accept universals.

In geometry, for example, when we wish to prove something about all triangles, we draw a particular triangle and reason about it, taking care not to use any characteristic which it does not share with other triangles. The beginner, in order to avoid error, often finds it useful to draw several triangles, as unlike each other as possible, in order to make sure that his reasoning is equally applicable to all of them. But a difficulty emerges as soon as we ask ourselves how we know that a thing is white or a triangle. If we wish to avoid the universals whiteness and triangularity, we shall choose some particular patch of white or some particular triangle, and say that anything is white or a triangle if it has the right sort of resemblance to our chosen particular. But then the resemblance required will have to be a universal.​

I wish I had better answers than I do, but if I had all the answers, this wouldn't be interesting . With post-modernism and social construction etc, people have been dissatisfied with analyticity all together and have simply denied that circles are necessarily round, so I suppose that's another option. There are some more serious semantic differences in that stance though, and I'm not a fan.

 

[apeiron]

Math is not a way of creating definite models - models are formal devices - real mathematics is a way of discovering our ideas of space, geometry and number- empirical observation can guide these discoveries but it is not the only source of guidance. What empirical model of reality would you say the theory of Riemann surfaces represents? How about the theory of differentiable structures on manifolds? Which empirical data did the Riemann hypothesis model? - what observations did it make "crisp"?
How about Thom's theory of cobordism of differentiable manifolds? After you explain all of these to me, you can move on to Chern-Simons invariants and then rational homotopy theory/ Oh yeah and maybe you could help me out with which empirical data the theory of Bieberbach groups was designed to model.


'

I appreciate the fact you like a good argument but you have to at least make an attempt to understand what the other side is saying. Saying you find things meaningless, or you don't believe, is not a very interesting position to hold. So I'll limit my responses.

"real mathematics is a way of discovering our ideas of space, geometry and number"

You mean pattern and form. Self-consistent organisation.

"What empirical model of reality would you say the theory of represents?"

As you say, it all starts with observation of the world - modelling at its most natural level. The ancients noted the regularities of the world and generalised to create some structural truths about objects like triangles and morphisms like spirals.

Then mathematics has developed by even greater generalisation - as articulated by category theory for example. And demonstrated in the move from euclidean to non euclidean geometry, or simple to complex number. We find that we exist in a world that is highly constrained (3D and flat - and scaled) and then generalise by successively removing those constraints to discover if self-consistent regularity still exists.

And you can see what happens when you do this with quarternions and octonions. The regularity frays. Properties like division erode as you generalise the dimensionality. Instead of producing crisp algebraic answers, the meaning of the algebra becomes vague.

So the most general maths can cease to model crisp properties that were there in the original "empirical" view.

When this happens, many say maths has just wandered off into the wilderness - or a landscape in string theory's case.

But my philosophical approach is different. I am saying that generalisations lead back to vague potential. And the way to rescue the situation is by also building the global constraints - the selection rules that represent the idea of "self-consistency" - back into the maths explicitly. So maths with scale.

"Science develops because people question or ideas of reality not because we model it."

Ideas are models - ideas formalised.