Why does mathematics works?

[Speakpigeon]

Why does mathematics works?

All is said in Wikipedia's article on Mathematics:

Through the use of abstraction and logic, mathematics developed from counting, calculation, measurement, and the systematic study of the shapes and motions of physical objects. Practical mathematics has been a human activity from as far back as written records exist. The research required to solve mathematical problems can take years or even centuries of sustained inquiry. Rigorous arguments first appeared in Greek mathematics, most notably in Euclid's Elements. Since the pioneering work of Giuseppe Peano (1858–1932), David Hilbert (1862–1943), and others on axiomatic systems in the late 19th century, it has become customary to view mathematical research as establishing truth by rigorous deduction from appropriately chosen axioms and definitions.
https://en.wikipedia.org/wiki/Mathematics

Mathematics works because, and only to the extent that, it is logical.

We got our logic through natural selection so, presumably, it was thoroughly tested over something like the 525 million years of the evolution of neuronal systems over the entire biosphere.

This doesn't mean that it should therefore work in all situations, only that it could be difficult for us to find one where it doesn't work.

Mathematicians can also invent theories that don't "work" because it just happens that there is nothing in the universe that works like that.

When a mathematical theory works, it can be thought of as a model of something real. For any such mathematical model, there is no good reason to claim that we know that it will work for ever, as if it was somehow a perfect model. In effect, we may believe that it will work for ever when in fact it won't because at some point in the future the model will be falsified by new facts. And we don't know the future.

In this case, we just don't know when it will stop to work. So, we can only believe that mathematical models will work. And then, that a model works doesn't mean that it is correct. Newton's laws of gravitation worked beautifully but then were effectively falsified by the more precise observation of Mercury's orbit.

Thus, we don't really know whether mathematics works since we don't know if it works for things we haven't been able to observe yet.

It may well be that we won't find anything ever for which mathematics doesn't work. However, this should be no surprise. I don't know of anything in nature that would somehow be illogical. So, again, as long as mathematics is logical, we should be safe.

This isn't specific to mathematics either. Language works, too, at least as long as you keep it logical. Any model works, as long as it is logical.

For example, you can try to think of Russian dolls. No mathematics. No language. Just your mind's power of imagination. Think of three dolls: doll A, doll B, doll C. Try to imagine a situation where doll A would be inside doll B and doll B inside doll C, while doll A wouldn't be inside doll C. Me, I can't. Our mind seems a pretty good model of reality and this before any mathematics at all.

So, the question of why mathematics works is trivial. It works because human logic works, and mathematics works only to the extent that it is logical.

The reason that logic works is less trivial. It works because it has been thoroughly tested by nature itself and finding a flaw in it is probably not easy at all. It seems safe to believe that finding a flaw in logic is beyond our current technological powers and will remain so for a very long time.

However, here too, there is no eternal guaranty. Only the future will tell.

I don't think I need to dwell on the question of the role played by abstraction in mathematics.
EB

 

[fromderinside]

Why does mathematics works?

All is said in Wikipedia's article on Mathematics:

Through the use of abstraction and logic, mathematics developed from counting, calculation, measurement, and the systematic study of the shapes and motions of physical objects. Practical mathematics has been a human activity from as far back as written records exist. The research required to solve mathematical problems can take years or even centuries of sustained inquiry. Rigorous arguments first appeared in Greek mathematics, most notably in Euclid's Elements. Since the pioneering work of Giuseppe Peano (1858–1932), David Hilbert (1862–1943), and others on axiomatic systems in the late 19th century, it has become customary to view mathematical research as establishing truth by rigorous deduction from appropriately chosen axioms and definitions.
https://en.wikipedia.org/wiki/Mathematics

Mathematics works because, and only to the extent that, it is logical.

...

EB

Mathematics works because it relates to physical world in an operational way. A cubit is a cubit and a knot is a knot wherever and they are applied and always relate to object treated operationally. Mathematics actually serves as the inspiration for logic. Logic is a discipline that often wraps up itself in irrational outcomes. Logic can apply to an imaginary world with little or no benefit to those who apply it beyond being some sort of gotcha game. Applying mathematics as a logical tool breaks the link between discipline and the real world.

Put the cart behind the horse sir.

As for the link between logic and natural selection you have no evidence for such. Natural selection is always material, treatable by systems linking the physical to observational tools - mathematics. Logic is not. For instance logic would tell us the next coin flip with be either different or the same as the previous for some personal causal reason (contradiction of terms). On the other hand mathematics shows us that coins will tend to fall equally heads or tails with just a skosh of edge probability as the result of observation.

 

[fast]

Effectively falsified?

It’s a Law!

 

[fromderinside]

Damn electron microscope and MR techniques.

Not a law a solution that works for what was known at the time. The theory has been extended by such as mercury observations and light bending showing that energy was better than force as a measure. The relationship between force and energy is

Work done is equal to force * distance. And, energy is equal to work done in unit time, which is equal to force*diatance/time, which is equal to force * velocity.

e=Fv

 

[steve_bank]

HeeHeee.what does he think it was based on. illogic? Tarot Cards, reading goat entrails?

Reasoning and logic are the same for all. Creationists use logic and draw technically valid conclusions, conclusion follows premise.

The difference between science and creationism is the set of premises which in science are usually based in part at least in experimental observation.

Math is based in logic, well no shit Sherlock. What does that have to do with the price of eggs?

Look at formal logic. You have AND, OR, and NOT(negation). From a basic principle in Boolean Algebra all logical expressions can be written as combinations of AND, OR, NOT.

Look at your own reasoning.

and
or
not
if then else

Everybody uses the same logic. It is inescapable. Formal logic is a symbolic structured form of common informal logic.

The problem is one of synthesis. Simply deriving premises and drawing a conclusion such as a mathematical proof works sometimes, a lot of times it does not. It is part experience and intuition from experience, part trial and error, and part logic. The trial and error part is now compurterized.

In the early 20th century Hilbert posed the question, are all mathematical truths provable, and is there an algorithm to find all mathematical truths. The question in part was the genesis of the Turing Machine, an algorithmic general purpose engine.

Göedel postulated in any logically consistent system, no ambiguities, there will be truths not provable in the system.

Aristotle thought given first principles he could logically deduce the universe, he was wrong. Linear logic is not a universal solution. The numerous syllogistic problems and fallacies demonstrate that.

P1 math is based on logic
C therfore math works becuase it is logical.

Conclusion does not follow from premise. Why does logic work, same quetion. The anwer is who knows. It is a combination of our brains plus evolution of thought passed on through culrure IMO.

Both logic and math in the end are empiracly validated by suage over time.

If I say ‘if it is raining and I am outside and I have no cover I will get wet’ why is it true?

A True if I am outside
B It is raining
C True if I am noit under cover
Dtrue if I am wet

Why is this logic true, why is logic true? How does logic 'work'?

If ( A & B & C ) then D

 

[steve_bank]

Damn electron microscope and MR techniques.

Not a law a solution that works for what was known at the time. The theory has been extended by such as mercury observations and light bending showing that energy was better than force as a measure. The relationship between force and energy is

Work done is equal to force * distance. And, energy is equal to work done in unit time, which is equal to force*diatance/time, which is equal to force * velocity.

e=Fv

Not to be pedantic but energy = E, e for me usually refers to the electron unit charge or voltage. Just sayin

 

[fromderinside]

Gotta be true. I was tired when I wrote my comment. So E's to you.

 

[Speakpigeon]

Effectively falsified?

It’s a Law!

Laws can be changed.

And what else could you possibly falsify?

Facts? But facts themselves are laws. Each fact is a law whose scope is limited to the fact itself: A implies A. (Thank you to have helped me think of that one! ).

Laws can be changed. You only need to have a majority, as Boris Johnson is being taught the hard way at the moment, to the enjoyment of the whole of Europe.
EB

 

[fromderinside]

Well, on lawfulness being changed I find that to be so. Until about a year ago I agreed that hearing movement was just a function of spectral detectability. Then in a swoop I realized that results I found back in 1976 meant the whole hearing sensing mechanism was involved in finding moving objects as the result of our ability to detect changes in tone as sounds moved through space near us, to hear doppler effects. Yes. The facts were the same but understanding of mechanisms changed in how they evolved to process them. A implies A changed to A implies B.

I won't bore you with why because this is not the place for that. But, if you are interested consider outer hair cell function both in bringing linearity to perception and to detecting correlated minute changes in acoustic activity.

As for facts being changed by perceptions of many you should consider GBs history rather than the discomfort they feel in making it. You are wrong on your perception of change.

 

[Speakpigeon]

As for facts being changed by perceptions of many you should consider GBs history rather than the discomfort they feel in making it. You are wrong on your perception of change.

You don't know what you are talking about.
EB

 

[fromderinside]

Wow. The MC was an agreement between King and lords not the people of GB.

As for current events: Is Commons going to vote to remain with CM? They're not going on 5 week time off before Queen's speech? Snap elections will take place? GB won't exit CM without a deal?

Actually I think the CM will extend parting just to keep from having to see their economies stressed by a decline in business because of a British recession.

You tell me what is and what will change. I think you can sense where I stand here by the questions I posed.

No sir, it's you who don't know what You're talking about. Few in Parliament, or GB for that matter, care what those in France think beyond whether to change the names of Brittany and Normandy. British folk want to leave the CM because of Germans pushing CM on immigration into a declining european workforce. Rather be racist than employed apparently.

People in Europe, including Britain, should read, understand and adhere to the economics of Paul Krugman.

Personally I think CM countries should require at least three language competency, one of being English the language of commerce, and a two year assimilation study program for all immigrants. I'm saying this because I believe that the order of people speaking are English, Mandarin, Hindi, Spanish, French, Arabic.

With the majority of those entering being from either arabic speaking countries or minor language countries it seems a good buffer for CM countries to do these things to encourage assimilation.

 

[steve_bank]

As for facts being changed by perceptions of many you should consider GBs history rather than the discomfort they feel in making it. You are wrong on your perception of change.

You don't know what you are talking about.
EB

Maybe he does not know what he is talking about, but he is certainly talking about what he is knowing about.

Again, the word law in science does not mean something immutable. Newton's laws of motion work within a bounded space of conditions. Law simply means a model that is so well tested in usage within specified conditions that it is accepted as true within conditions without debate.

 

[fromderinside]

^^^ that.

 

[fast]

https://www.google.com/amp/s/www.li...-in-science-definition-of-scientific-law.html

 

[fromderinside]

^^^ too

 

[Speakpigeon]

https://www.google.com/amp/s/www.li...-in-science-definition-of-scientific-law.html

What Is a Law in Science?
Laws are descriptions — often mathematical descriptions — of natural phenomenon; for example, Newton's Law of Gravity. These laws simply describe the observation.

Do laws change?
Just because an idea becomes a law, doesn't mean that it can't be changed through scientific research in the future. The use of the word "law" by laymen and scientists differ. When most people talk about a law, they mean something that is absolute. A scientific law is much more flexible. It can have exceptions, be proven wrong or evolve over time, according to the University of California.

"A good scientist is one who always asks the question, 'How can I show myself wrong?'" Coppinger said. "In regards to the Law of Gravity or the Law of Independent Assortment, continual testing and observations have 'tweaked' these laws. Exceptions have been found. For example, Newton's Law of Gravity breaks down when looking at the quantum (sub-atomic) level. Mendel's Law of Independent Assortment breaks down when traits are "linked" on the same chromosome."

"Laws are descriptions (...) of natural phenomenon".

"Just because an idea becomes a law, doesn't mean that it can't be changed through scientific research in the future".

Is that clear enough?

I rest my case.
EB

 

[fromderinside]

idea not equal description.

Case dismissed.

 

[fast]

A law undergoes immense scientific scrutiny. That’s not to say a law cannot be proven false, but in the case of Newton’s laws of motion, the scientific community has not stamped it as false; in fact, it’s still a law! We just recognize the scope of its applicability now.

 

[Speakpigeon]

A law undergoes immense scientific scrutiny. That’s not to say a law cannot be proven false, but in the case of Newton’s laws of motion, the scientific community has not stamped it as false; in fact, it’s still a law! We just recognize the scope of its applicability now.

That's sophistry and equivocation. Scientists following Newton thought of the law of gravitation as universal and correct. Mercury put a foot into that.

Whether scientists today keep referring to the same formula as a law is entirely irrelevant. This formula is no longer seen as the universal and correct formula to describe gravitation because this characterisation is now that of Einstein's relativist formula.

You may still call it a law but it's no longer the law that applies to the same thing. It was characterised as universal, that is to say as applying to the entire universe, and it is no longer because Relativity says that there are regions of the universe where Newton's formula doesn't apply. And in fact, the relativist formula will be better everywhere. Scientists are currently able to measure the inaccuracy of Newton's formula already in the Earth region.

Newton's law of universal gravitation
Newton's law of universal gravitation states that every particle attracts every other particle in the universe with a force which is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers.
https://en.wikipedia.org/wiki/Newton's_law_of_universal_gravitation

EB

 

[fromderinside]

What I grant you is that Newton's laws don't describe what we can measure as well as does the theory of general relativity. All that means is that the mechanics of the presently observable universe can be more precisely measured. That does not make the relations between Force, mass, and acceleration devised by Newton false. When scientists and engineers use Newtons theory to measure requirements and trajectories of devices traveling between bodies one cannot say the theory is false. Theory has come to be a process where one set of generalizations are superseded by another set generalizations when more information comes available. It is not a win lose proposition. In fact units of measure which are derived using Newton's laws are still set as standards of measurement for weight, mass, length and local time.

Mankind still uses systems of measurement dating to egyption times before any theory of relations between object and motion were derived. Not even those those measure are falsified.

You should back off from a philosophical principle that the advocate of that principle came to reject. It makes you look, well, stuck in a world where scientific method is frowned upon.

 Scientific_method

The scientific method is an empirical method of acquiring knowledge that has characterized the development of science since at least the 17th century. It involves careful observation, applying rigorous skepticism about what is observed, given that cognitive assumptions can distort how one interprets the observation. It involves formulating hypotheses, via induction, based on such observations; experimental and measurement-based testing of deductions drawn from the hypotheses; and refinement (or elimination) of the hypotheses based on the experimental findings.

Theories are targets for experimental test. And as theories come to comver more information it will be superceded by the more powerful theory. That does not mean previous theory is wrong. Usually it means there is more infomatin that can be explained by another formulation.