Mathematics versus Science

[EmptyHand]

There is a very important difference between mathematics and science. The goal of mathematics is PROOF whereas the goal of science is REFUTATION (disproof).

No scientific theory is ever proven. Every scientific theory and every scientific model ever created has a limited domain of validity. Beyond these limits the validity and predictive accuracy of the model breaks down. Scientists seek to find these limits and by doing this they seek examples where a theory breaks down, i.e. is refuted. For example, classical physics breaks down in the domain of the very small (where quantum mechanics is required) and in the domain of the very large (where general relativity is required).

Using the word "proof" in regard to a scientific theory is a significant conceptual misunderstanding. For further reading on these matters, one can read the works of Karl Popper and Thomas Kuhn.

"Proof" should only be spoken about in the domain of (pure) mathematics. This post is NOT intended to be anti-science. Both science and mathematics are MARVELOUS. But this important distinction is often overlooked and leads to unnecessary disputes.


eH

 

[Infundibulum]

Interesting that you make the distinction between science and mathematics. Generally because demonstrating that something is something is relatively easy but demonstrating that this something isn't something else is the tough, there is not ultimate proof in anything. I also have the feeling that when you get an ultimate proof with mathematics this proof is too vague to be of any use. Would you like to paint your position with examples?

Using the word "proof" in regard to a scientific theory is a significant conceptual misunderstanding. For further reading on these matters, one can read the works of Karl Popper and Thomas Kuhn.

Can you also further expand on this? Obviously, you do not expect us to read a book to discuss in this thread.

 

[spinCycle]

Very little scientific refutation could take place without math.

 

[EmptyHand]

I also have the feeling that when you get an ultimate proof with mathematics this proof is too vague to be of any use. Would you like to paint your position with examples?

Well, the usefulness of mathematical proof ranges from the Euclidean geometry which you probably encountered in high school all the way to proofs of error bounds for numerical computations. These are "absolute" proofs which are also useful in the physical world.

Obviously, you do not expect us to read a book to discuss in this thread.

For starters, here is a wiki entry on "falsifiability" which Popper championed:

Falsifiability - Wikipedia

en.wikipedia.org

Very little scientific refutation could take place without math.

Indeed. Math is an essential tool for the advancement of science. I'm not favoring one discipline over the other. Just pointing out a difference which is often overlooked.

eH

 

[Infundibulum]

I also have the feeling that when you get an ultimate proof with mathematics this proof is too vague to be of any use. Would you like to paint your position with examples?

Well, the usefulness of mathematical proof ranges from the Euclidean geometry which you probably encountered in high school all the way to proofs of error bounds for numerical computations. These are "absolute" proofs which are also useful in the physical world

But could you give us particular examples? Euclidean geometry proofs are based on axioms that cannot themselves be proven or demonstrated, such as that parallel lines do not intersect. I know nothing about proofs of error bounds for numerical computations or what it means.

There must be a particular example in lay terms where a mathematics gave absolute proof.

Obviously, you do not expect us to read a book to discuss in this thread.

For starters, here is a wiki entry on "falsifiability" which Popper championed:

Falsifiability - Wikipedia

en.wikipedia.org

My apologies but I'll go out on a limb here, it is bad manners to ask people to acquaint themselves with further reading in the concepts you're discussing. Your OP should be self-contained and, ideally, to lay the concepts you wish to discuss in your own word without relying on the words of others to explain what you should be explaining yourself only.

 

[EmptyHand]

Infundibulum,

I apologize for not properly answering your questions. For your first point, any proof from geometry is indeed "absolute", if I am using the word properly in your context. It is true that the proof rests on assumed axioms but given the axioms the result follows indubitably and is not "falsifiable" in the way that a physical theory is falsifiable. In applications, sometimes the axioms do not hold and therefore the result is false. For example, geometry on a sphere (an example of a non-euclidean geometry) has NO parallel lines. However in the domain of pure mathematics, we are not concerned with "applications". A proof is absolute but an application is not. Applications take us out of (pure) mathematics and into the domain of science.

And I apologize for referencing outside sources. I'm afraid I cannot make this thread self-contained as it touches on the deep waters of the philosophy of science and the philosophy of math. I can only hope to make a fairly cogent point and indicate directions for followup.

eH

 

[InMotion]

A very nice read about the separations of reality, spiritual experiences, science, and mathematics is "Hegel's Phenemology of the Spirit". There is an e-book copy on this website, here: Hegel's Phenomenology of Spirit - Spirituality & Mysticism - Welcome to the DMT-Nexus.

I find it interesting you think science is about 'disproof' when in reality almost all published works are founded on evidence. Proof is a strange word that doesn't translate well from mathematics to science. Of course once a repeatable finding is noted that contradicts a current theory, it means the theory needs to be adapted or dismantled. However, that is not necessarily the goal of science itself, and still as such the existence of the out-lier must be founded with evidence.

Guess we're on the same page just looking at the same things from different directions.

 

[Aetherius Rimor]

Science "proves" concepts to a degree of certainty, rather than absolute proofs.

Overwhelming evidence in support of an idea, and no evidence refuting an idea, is the equivalent of a proof in science.

If you perform an experiment 999,999,999 times and obtain the same result every time, you have not proven that a different result is impossible. Just with all probability highly unlikely. Even if the answer is different on the trillionth time, you have proven your theory as false, but the theory is still reliable for most needs.

Which is why euclidean geometry, and classical mechanics are still necessary and useful.

 

[nen888]

EmptyHand wrote:

any proof from geometry is indeed "absolute"

..thank you for this thread, i enjoy the world of math, while not being a mathematician by any stretch..
..from a pure maths POV, i think some are missing some of the interesting differences with science, but which science could not, given history, happily dispense with..

from a mathematician discussion: Mathematical Intuition—What Is It?

Proving theorems is not mechanical; proving theorems does require formal manipulation. Yet proving theorems also requires the use of intuition, the ability to see what is reasonable or not, and the ability to put all these together. Blindly using a lemma from even the most famous textbook can be dangerous, as the story shows.

Gödel wrote:

Geometrical intuition, strictly speaking, is not mathematical, but rather a priori physical, intuition. In its purely mathematical aspect our Euclidean space intuition is perfectly correct, namely it represents correctly a certain structure existing in the realm of mathematical objects. Even physically it is correct 'in the small'.

[cited in
INTUITIONS OF THREE KINDS IN GÖDEL'S VIEWS ON THE CONTINUUM, Princeton Uni. paper]

..the Hypercube was first 'seen' by Hinton in a 'mental space' before being proved..
Rudy Rucker wrote (in Infinity and the Mind):

..the human mind is incapable of formulating (or mechanizing) all of it's math..

.
.

 

[The Neural]

Science "proves" concepts to a degree of certainty, rather than absolute proofs.

Overwhelming evidence in support of an idea, and no evidence refuting an idea, is the equivalent of a proof in science.

If you perform an experiment 999,999,999 times and obtain the same result every time, you have not proven that a different result is impossible. Just with all probability highly unlikely. Even if the answer is different on the trillionth time, you have proven your theory as false, but the theory is still reliable for most needs.

This was the shortest, yet most comprehensible description of science's perspective on the concept of "proof in science".

:thumb_up:

 

[EmptyHand]

Science "proves" concepts to a degree of certainty, rather than absolute proofs.

Overwhelming evidence in support of an idea, and no evidence refuting an idea, is the equivalent of a proof in science.

"Degree of certainty" is the essence of the notion of "probability." But scientific theories are not "probably true" or "probably false." They have predictive accuracy within delineated boundaries of applicability. All scientific theories have predictive limits and thus regions of experience where they break down. Investigating the breakdown regions is the most exciting science to practice. "Falsifiability" is the key notion:

Falsifiability - Wikipedia

en.wikipedia.org

eH

 

[InfiniteFacticity]

A very nice read about the separations of reality, spiritual experiences, science, and mathematics is "Hegel's Phenemology of the Spirit". There is an e-book copy on this website, here: Hegel's Phenomenology of Spirit - Spirituality & Mysticism - Welcome to the DMT-Nexus.

This link sends me to a page that says "You have tried to enter a area where you didn't have access."

Although, perhaps the Nexus knows that the last time I picked up the Phenomenology of Spirit I got two pages into the preface and nearly had an aneurysm, so maybe it's just being a smart-ass.

On topic, my understanding is that mathematics has nothing to do with empirical evidence. Mathematicians may take inspiration from measurable and observable aspects of the world, but mathematical proofs do not necessarily need to even attempt to model reality. Science includes math as a loose, or good enough model of certain phenomena, but the formulas, algorithms, etc are never precisely correlated with observations.

As far as "proof" usage goes, I agree that it makes more sense to use only in math, and I am fairly sure that every scientist would agree.

 

[EmptyHand]

For a slightly nuanced view of the "absolute" nature of mathematics expressed in previous posts, the book "Proofs and Refutations" is highly recommended.

Here is a description on Wikipedia:

Proofs and Refutations - Wikipedia

en.wikipedia.org

 

[GARMONIUM]

Mathematics is a common language throughout the universe (in all evolved alien civilisations they should know the same math that we do and I am sure some more) , the same should be the case with fundamental science but math to me seems to be like the source code to the rest of reality so could be seen as primal. Thats just my opinion

Slightly off topic: Do any of you guys know of anyone who can virtually model - develop and program microprocessors and microcontrollers. 8)

 

[EmptyHand]

Slightly off topic: Do any of you guys know of anyone who can virtually model - develop and program microprocessors and microcontrollers. 8)

Do you mean program actual microcontrollers or virtual microcontrollers? To learn how to program actual microcontrollers I suggest buying and experimenting with the Arduino. It costs around $25 on Amazon and "starter kits" with extra electronic components are available for a bit more. Here is the main website:

http://www.arduino.cc/

I'm not exactly sure what you mean by "virtually" model a microcontroller but if you mean simulate a microcontroller in software, here is an Arduino simulator:

http://www.arduino.com.au/Simulator-for-Arduino.php

Learning to control hardware through software is not that difficult and is quite fun with Arduino. I'm experimenting with making devices to assist in learning meditation using the Arduino.

eH

 

[GARMONIUM]

Thank you for this info EmptyHand looks interesting

When I said virtually model I meant ( create virtual prototypes in software like this : NI Multisim Power Pro Product ) and in terms of programming I was referring to making physical prototypes based on standard microchips programmed ( burned ) with specific new types of function sets.

 

[EmptyHand]

Here is a link to a brand new article, "Philosophy and the Practice of Bayesian Statistics", that is a bit technical in spots but is an excellent overview of the modern philosophy of science.

http://onlinelibrary.wiley.com/doi/10.1111/j.2044-8317.2011.02037.x/abstract

Section 6 is especially relevant to the discussion in this thread regarding the nature and practice of science.

eH

 

[hixidom]

I disagree with the general statement that the goal of mathematics is proof and the goal of science is disproof. Disproof is equally common in mathematics, and I don't think any scientist would say "I study what reality isn't", which may be partly true, but it seems to me that you're putting a spin on something that is ultimately equivalent to its opposite (proof).

If there is a difference, I would say it is that mathematicians develop complete and absolute representations of hypothetical realities while scientists develop hypothetical models in order to explain perceptual observations.

 

[primordium]

Actually, the philosophy of mathematics is an expansive and growing field. One school, fictionalism, actually doubts the ontological validity of mathematical objects. Moreover, other philosophers like Quine or Putnam, have put forth arguments that subsume mathematics to empiricism (i.e., placing mathematics and science on equal footing).

Of course, I'm much more of a Pythagorean and consider mathematics one of the surest ways to affirm incorporeality.

 

[primordium]

If there is a difference, I would say it is that mathematicians develop complete and absolute representations of hypothetical realities while scientists develop hypothetical models in order to explain perceptual observations.

Hypothetical? Max Tegmark might disagree!