Interestingly enough, both philosophy and science operate with axioms.

Only mathematics uses axioms, science does not.

Also, I find the argument amusing that something cannot be proven in reality unless it is falsifiable, when the same line of thought usually claims that (1) something is only falsifiable according to logic (propositions), and (2) logic is not connected to reality.

It's not so difficult: logic itself doesn't tell us anything about reality (you can imagine perfectly logical worlds that don't exist in reality), but that doesn't mean that you cannot use logic in interpreting empirical evidence. Logic is here used as a means of interpreting your data, but it doesn't constitute data itself. You can falsify a theory with experimental evidence and logical reasoning, but you cannot falsify logic itself. In other words, logic itself doesn't tell us anything about the real world, but it can be used in a description of the real world, but also of imaginary worlds.

I have always had a problem with this, er... logic...

Congress sets out to design a horse and ends up with a camel, indicating the incompetence of Congress. This is a quip, indicating that Government and its agents are not particularly clever or competent. Which is not surprising. Since the Government is not run like a business there is no incentive or imperative to function well. All the persons involved are going to get paid regardless of the outcome.

Congress sets out to design a horse and ends up with a camel, indicating the incompetence of Congress. This is a quip, indicating that Government and its agents are not particularly clever or competent. Which is not surprising. Since the Government is not run like a business there is no incentive or imperative to function well. All the persons involved are going to get paid regardless of the outcome.

Ba'al Chatzaf

Most businesses don't run well either despite the incentive, hence bankruptcy. Many people don't understand that business failures are an important part of the capitlaist system.

Can you show me some science, any science at all, that does not include math as an essential component?

If not, well, your pronouncement has a problem.

Not at all. There are physical theories and physical laws, but no physical axioms. That those theories make use of mathematics which in their turn are based on a set of axioms is not relevant. For the physicist the mathematics is just a convenient tool and in general they don't bother too much about the fundamentals. A good example is the Dirac delta function, invented by a physicist. It's a very useful tool, but it didn't have a real mathematical basis (in fact it isn't a function at all) until the mathematician Laurent Schwartz developed his theory of distributions in which the Dirac "function" can be rigorously defined. For the physicists it doesn't make a difference, the axioms of distribution theory are not essential to the physical theories. There are more examples of (quasi-)mathematical tools invented by physicists that only later got a solid foundation by the work of mathematicians.

Can you show me some science, any science at all, that does not include math as an essential component?

If not, well, your pronouncement has a problem.

Not at all. There are physical theories and physical laws, but no physical axioms. That those theories make use of mathematics which in their turn are based on a set of axioms is not relevant. For the physicist the mathematics is just a convenient tool and in general they don't bother too much about the fundamentals. A good example is the Dirac delta function, invented by a physicist. It's a very useful tool, but it didn't have a real mathematical basis (in fact it isn't a function at all) until the mathematician Laurent Schwartz developed his theory of distributions in which the Dirac "function" can be rigorously defined. For the physicists it doesn't make a difference, the axioms of distribution theory are not essential to the physical theories. There are more examples of (quasi-)mathematical tools invented by physicists that only later got a solid foundation by the work of mathematicians.

To amplify your remark, Newton invented the technique of fluxions (calculus where time is the variable). Newtons calculus of fluxions was invented to express motions, accelerations and relate velocities and accelerations to position yield up a theory of differential equations. Both Newton and Liebniz invoked infinitesimals, quantities that are not zero yet smaller than any definite quantity. At the time this made no mathematical sense, but it worked. This boldness was a mathematical scandal until the early 19th century when August Cauchy and others invented the rigorous theory of limits to make good on Newton's and Leibniz frontal assault on logic. It was until the late 1950's that Abraham Robinson rigorously grounded the theory of infinitesimal quantities. That is over 250 years after the fact.

Both Newton and Leibniz created a mathematical tool that worked magnificently and finally captured the concept of motion properly, but the tool had no basis in logic at the time it was invented. It just worked.

That those theories make use of mathematics which in their turn are based on a set of axioms is not relevant.

Dragonfly,

Then you should have no problem at all showing me some science that does not use math. What do you suggest?

After all, according to this line of thinking, math has no connection to reality and is only a "useful tool," so science at root doesn't really need it.

How come examples of science without math are so hard to come by?

That those theories make use of mathematics which in their turn are based on a set of axioms is not relevant.

Dragonfly,

Then you should have no problem at all showing me some science that does not use math. What do you suggest?

After all, according to this line of thinking, math has no connection to reality and is only a "useful tool," so science at root doesn't really need it.

How come examples of science without math are so hard to come by?

Michael

Not only useful, but necessary. Physics without its mathematical tool would end up looking like Aristotle's nonsense.

Physics without its mathematical tool would end up looking like Aristotle's nonsense.

Bob,

Then science (at least physics from your statement) includes axioms. Otherwise it looks like nonsense. Correct?

btw - On looking up the Dirac delta function, I saw that it was conceived as ordinal measurement, not cardinal measurement, which is still math.

Michael

All of quantitative science is grounded on assumptions. But they are not considered (by scientists) to be self evident truth. All sciences have to have a starting point, but the grundlagen need not be axiomatic.

Physics without its mathematical tool would end up looking like Aristotle's nonsense.

Bob,

Then science (at least physics from your statement) includes axioms. Otherwise it looks like nonsense. Correct?

btw - On looking up the Dirac delta function, I saw that it was conceived as ordinal measurement, not cardinal measurement, which is still math.

Michael

Take a course in Lebesque Integration before making pronouncements.

The Dirac Delta Function is defined as a distribution operator on Lesbesque Measurable functions. You can't get any more cardinal than that. Anything that appears to the right of an integral sign is a cardinal not an ordinal. All of the theory of real and complex functions are based on cardinals.

Physics without its mathematical tool would end up looking like Aristotle's nonsense.

Bob,

Then science (at least physics from your statement) includes axioms. Otherwise it looks like nonsense. Correct?

No. Bob gave you just the example of Newton who used a mathematical tool that worked fine for centuries and only much later was put on a rigorous basis with axioms. Do you really think that Newton's theory only became valid when his mathematical tool was made mathematically rigorous centuries later? It may be nice to put the dots on the i's and cross the t's, but it isn't essential to the content of a text.

btw - On looking up the Dirac delta function, I saw that it was conceived as ordinal measurement, not cardinal measurement, which is still math.

Huh? Where did you get that notion? Are you perhaps confusing the mathematical notion of "measure" (see for example here) with "measurement"? Anyway, your point is moot, as the notion of the Dirac delta as a measure is just part of the formal theory of distributions that was developed later, and was not used for the practical tool that Dirac invented.

You guys use big words (I believe for intimidation) and talk around the issue, but I still don't see any science without math.

Every bit of what I have seen so far in science under the heading of "empirical" includes math to get there. So either math is fundamental, or it is merely a "useful tool," which means it can be eliminated.

Math uses axioms. If something cannot work without using math, it must use the same axioms.

Until I see science that is "empirical" that does not use math, I see no need to try to deny the obvious. But do carry on.

The moment you have "this comes before that," you have first and second, i.e. ordinal measurement. This is a quite simple concept and there is no need to hairsplit to understand it. And hairsplitting certainly will not push it out of existence.

Can you show me some science, any science at all, that does not include math as an essential component?

If not, well, your pronouncement has a problem.

Michael

There is a slight problem with the statement, it should read "Only pure mathematics uses axioms, science does not." Micheal, you keep bringing up this point and I wish I could make you see the difference. When a scientist uses any kind of mathematics he is attaching a physical meaning to the symbols whereas the mathematician is not. This is a huge difference. When we attach a physical meaning then we get to apply some measure of "truth" or "false" to the relation but in pure mathematics we are only concerned with consistency. So science utilizes mathematics (your use of 'include' is somewhat ambiguous) and sometimes it even precedes mathematics as the above example illustrates, but science is a fundamentally different activity.

The moment you have "this comes before that," you have first and second, i.e. ordinal measurement. This is a quite simple concept and there is no need to hairsplit to understand it. And hairsplitting certainly will not push it out of existence.

Sorry, this goes completely over my head, I've no idea what this has to do with the Dirac delta.

OK, the current in an electrical circuit is related to the voltage and the resistance.

GS,

Do you mean to say that electrical circuits were built without math? And that understanding the relation between voltage and resistance was developed without math? Or even understanding voltage and resistance?

The moment you have "this comes before that," you have first and second, i.e. ordinal measurement. This is a quite simple concept and there is no need to hairsplit to understand it. And hairsplitting certainly will not push it out of existence.

Sorry, this goes completely over my head, I've no idea what this has to do with the Dirac delta.

Dragonfly,

As I understand it, the Dirac delta is not a random distribution, but a distribution that occurs in sequence. Where you have sequence, you have first, second, third, etc.

The moment you have "this comes before that," you have first and second, i.e. ordinal measurement. This is a quite simple concept and there is no need to hairsplit to understand it. And hairsplitting certainly will not push it out of existence.

Sorry, this goes completely over my head, I've no idea what this has to do with the Dirac delta.

Dragonfly,

As I understand it, the Dirac delta is not a random distribution, but a distribution that occurs in sequence. Where you have sequence, you have first, second, third, etc.

Ergo, math.

Michael

Rats, all my systems engineering books discussing the Dirac-Delta function are in transit or else I'd be all over this discussion :-).

As I understand it, the Dirac delta is not a random distribution, but a distribution that occurs in sequence. Where you have sequence, you have first, second, third, etc.

In mathematics commonly used terms often have a special meaning. I'm talking here about the mathematical notion of distribution, see for example here. Anyway, your argument doesn't hold, as the definition of the Dirac delta as a distribution is just an example of a later formalization of the inexact but effective tool that was used by Dirac and all physicists after him. The formal aspect of mathematics aren't really relevant to the correctness of physical theories, just as the dots on the i's are not essential to the correctness of the content of a text.

Do you mean to say that electrical circuits were built without math? And that understanding the relation between voltage and resistance was developed without math? Or even understanding voltage and resistance?

Michael

Electrical circuits were around before mankind and his mathematics. One can postulate a relationship without quantifying it. The measurements we make serve to make our understanding of relationships more accurate.

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## Dragonfly

Only mathematics uses axioms, science does not.

It's not so difficult: logic itself doesn't tell us anything about reality (you can imagine perfectly logical worlds that don't exist in reality), but that doesn't mean that you cannot use logic in interpreting empirical evidence. Logic is here used as a

meansof interpreting your data, but it doesn't constitute data itself. You can falsify a theory with experimental evidence and logical reasoning, but you cannot falsify logic itself. In other words, logic itself doesn't tell us anything about the real world, but it can be used in a description of the real world, but also of imaginary worlds.Perhaps one day you'll understand it...

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## BaalChatzaf

Congress sets out to design a horse and ends up with a camel, indicating the incompetence of Congress. This is a quip, indicating that Government and its agents are not particularly clever or competent. Which is not surprising. Since the Government is not run like a business there is no incentive or imperative to function well. All the persons involved are going to get paid regardless of the outcome.

Ba'al Chatzaf

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## James Heaps-Nelson

Most businesses don't run well either despite the incentive, hence bankruptcy. Many people don't understand that business failures are an important part of the capitlaist system.

Jim

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## Michael Stuart Kelly

Dragonfly,

If you say so.

Can you show me some science, any science at all, that does not include math as an essential component?

If not, well, your pronouncement has a problem.

Michael

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## Dragonfly

Not at all. There are physical theories and physical laws, but no physical axioms. That those theories make use of mathematics which in their turn are based on a set of axioms is not relevant. For the physicist the mathematics is just a convenient tool and in general they don't bother too much about the fundamentals. A good example is the Dirac delta function, invented by a physicist. It's a very useful tool, but it didn't have a real mathematical basis (in fact it isn't a function at all) until the mathematician Laurent Schwartz developed his theory of distributions in which the Dirac "function" can be rigorously defined. For the physicists it doesn't make a difference, the axioms of distribution theory are not essential to the physical theories. There are more examples of (quasi-)mathematical tools invented by physicists that only later got a solid foundation by the work of mathematicians.

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## BaalChatzaf

To amplify your remark, Newton invented the technique of fluxions (calculus where time is the variable). Newtons calculus of fluxions was invented to express motions, accelerations and relate velocities and accelerations to position yield up a theory of differential equations. Both Newton and Liebniz invoked

infinitesimals, quantities that are not zero yet smaller than any definite quantity. At the time this made no mathematical sense, but itworked. This boldness was a mathematical scandal until the early 19th century when August Cauchy and others invented the rigorous theory of limits to make good on Newton's and Leibniz frontal assault on logic. It was until the late 1950's that Abraham Robinson rigorously grounded the theory of infinitesimal quantities. That is over 250 years after the fact.Both Newton and Leibniz created a mathematical tool that worked magnificently and finally captured the concept of motion properly, but the tool had no basis in logic at the time it was invented. It just worked.

Ba'al Chatzaf

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## Michael Stuart Kelly

Dragonfly,

Then you should have no problem at all showing me some science that does not use math. What do you suggest?

After all, according to this line of thinking, math has no connection to reality and is only a "useful tool," so science at root doesn't really need it.

How come examples of science without math are so hard to come by?

Michael

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## BaalChatzaf

Not only useful, but necessary. Physics without its mathematical tool would end up looking like Aristotle's nonsense.

Ba'al Chatzaf

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## Michael Stuart Kelly

Bob,

Then science (at least physics from your statement) includes axioms. Otherwise it looks like nonsense. Correct?

btw - On looking up the

Dirac delta function, I saw that it was conceived as ordinal measurement, not cardinal measurement, which is still math.Michael

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## BaalChatzaf

All of quantitative science is grounded on assumptions. But they are not considered (by scientists) to be self evident truth. All sciences have to have a starting point, but the grundlagen need not be axiomatic.

Ba'al Chatzaf

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## BaalChatzaf

Take a course in Lebesque Integration before making pronouncements.

The Dirac Delta Function is defined as a distribution operator on Lesbesque Measurable functions. You can't get any more cardinal than that. Anything that appears to the right of an integral sign is a cardinal not an ordinal. All of the theory of real and complex functions are based on cardinals.

Ba'al Chatzaf

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## Dragonfly

No. Bob gave you just the example of Newton who used a mathematical tool that worked fine for centuries and only much later was put on a rigorous basis with axioms. Do you

reallythink that Newton's theory only became valid when his mathematical tool was made mathematically rigorous centuries later? It may be nice to put the dots on the i's and cross the t's, but it isn't essential to the content of a text.Huh? Where did you get that notion? Are you perhaps confusing the mathematical notion of "measure" (see for example here) with "measurement"? Anyway, your point is moot, as the notion of the Dirac delta as a measure is just part of the formal theory of distributions that was developed later, and was not used for the practical tool that Dirac invented.

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## Michael Stuart Kelly

You guys use big words (I believe for intimidation) and talk around the issue, but I still don't see any science without math.

Every bit of what I have seen so far in science under the heading of "empirical" includes math to get there. So either math is fundamental, or it is merely a "useful tool," which means it can be eliminated.

Math uses axioms. If something cannot work without using math, it must use the same axioms.

Until I see science that is "empirical" that does not use math, I see no need to try to deny the obvious. But do carry on.

Michael

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## Michael Stuart Kelly

Dragonfly,

The moment you have "this comes before that," you have first and second, i.e. ordinal measurement. This is a quite simple concept and there is no need to hairsplit to understand it. And hairsplitting certainly will not push it out of existence.

Michael

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## tjohnson

There is a slight problem with the statement, it should read "Only

mathematics uses axioms, science does not." Micheal, you keep bringing up this point and I wish I could make you see the difference. When a scientist uses any kind of mathematics he is attaching apurephysical meaningto the symbols whereas the mathematician is not. This is a huge difference. When we attach a physical meaning then we get to apply some measure of "truth" or "false" to the relation but in pure mathematics we are only concerned with consistency. So scienceutilizesmathematics (your use of 'include' is somewhat ambiguous) and sometimes it even precedes mathematics as the above example illustrates, but science is a fundamentally different activity.## Link to comment

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## Michael Stuart Kelly

GS,

Fine. Show me science without math. Any old science will do.

Michael

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## tjohnson

OK, the current in an electrical circuit is related to the voltage and the resistance.

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## tjohnson

Actually, I'm not even sure what is meant by 'axiom' here. I would say mathematics begins with

definitionsmore so than axioms.## Link to comment

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## Dragonfly

Sorry, this goes completely over my head, I've no idea what this has to do with the Dirac delta.

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## Michael Stuart Kelly

GS,

Do you mean to say that electrical circuits were built without math? And that understanding the relation between voltage and resistance was developed without math? Or even understanding voltage and resistance?

Michael

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## Michael Stuart Kelly

Dragonfly,

As I understand it, the Dirac delta is not a random distribution, but a distribution that occurs in sequence. Where you have sequence, you have first, second, third, etc.

Ergo, math.

Michael

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## James Heaps-Nelson

Rats, all my systems engineering books discussing the Dirac-Delta function are in transit or else I'd be all over this discussion :-).

Jim

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## Dragonfly

In mathematics commonly used terms often have a special meaning. I'm talking here about the

mathematicalnotion of distribution, see for example here. Anyway, your argument doesn't hold, as the definition of the Dirac delta as a distribution is just an example of a later formalization of the inexact but effective tool that was used by Dirac and all physicists after him. Theformalaspect of mathematics aren't really relevant to the correctness of physical theories, just as the dots on the i's are not essential to the correctness of the content of a text.## Link to comment

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## tjohnson

Electrical circuits were around before mankind and his mathematics. One can postulate a relationship without quantifying it. The measurements we make serve to make our understanding of relationships more accurate.

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