Mathematics, Discovered or Invented?

[Polymath257]

I wouldn't. I'm waiting for some believer to come up with a set of axioms and operations which describe a system of divinity that is as consistent as mathematics. That would show that the divine is a Platonic ideal.
But the reality is that there is no system to the madness. All the divine "axioms" are just rectally derived claims with no connection to each other, and usually contradictory.
Mathematical axioms are discovered through the "observation" of numbers and forms.

Even math cannot be proved to be consistent (once it is strong enough to model the natural numbers).

I would strongly object that the axioms of math are discovered. We might observe what happens with very small numbers (say, those less than a million) and use that as an intuitive basis for forming an axiom system. But that is *us* making up the axiom system to fit our intuitions, not something that is forced by our observations.

For example, there are axiom systems that allow every specific arithmetic statement (like 341+782=1223) to be proved, but where commutativity of addition cannot be proved. Such a minimal system is completely consistent with our observations, but even simply general results can fail.

 

[Shaul]

I wouldn't. I'm waiting for some believer to come up with a set of axioms and operations which describe a system of divinity that is as consistent as mathematics. That would show that the divine is a Platonic ideal.
But the reality is that there is no system to the madness. All the divine "axioms" are just rectally derived claims with no connection to each other, and usually contradictory.
Mathematical axioms are discovered through the "observation" of numbers and forms.

I can think of an inconsistent set of axioms in mathematics. In Euclidean Geometry there are axioms. Yet in non-Euclidean geometry those axioms are modified. They are consistent within each branch of Geometry but not across the branches.

 

[Polymath257]

I can think of an inconsistent set of axioms in mathematics. In Euclidean Geometry there are axioms. Yet in non-Euclidean geometry those axioms are modified. They are consistent within each branch of Geometry but not across the branches.

Such inconsistent sets of axioms are NOT considered to be about the same things, though. The 'objects' of study in Euclidean and non-Euclidean geometry are different geometries. In theology, the inconsistent axioms are supposed to be all about theology.

 

[Mock Turtle]

I have read one book as to the history of mathematics - can't remember which one but it was well worth reading even if I seem to have forgotten most of what was in the book. Still have it somewhere no doubt.

 

[Kfox]

I think that mathematics is discovered.

Where was it discovered?

One piece of evidence is that maths has been discovered multiple times independently.

When?

 

[Polymath257]

Where was it discovered?

When?

I think @Heyo was getting at the fact that it is common in math for the same idea to appear independently in different places and even at different times.

One example is the development of calculus, which is attributed to both Newton and Leibnitz (although others did contribute). Both came up with essentially the same techniques independently.

The 'where' of discovery seems to be of less interest, though, than the 'who' and 'when'. Of course, different concepts in math arose at different times. Even today it is common for two researchers to find essentially the same results and techniques even though they have no contact.

That does suggest 'discovery'.

 

[Jayhawker Soule]

I go with a mixture of discovery and invention.

I go with deduced.

 

[Polymath257]

I go with deduced.

We deduce by discovering a proof.

 

[Shaul]

Such inconsistent sets of axioms are NOT considered to be about the same things, though. The 'objects' of study in Euclidean and non-Euclidean geometry are different geometries. In theology, the inconsistent axioms are supposed to be all about theology.

So too there are likewise different domains within theology. So not accepting inconsistent axioms in theology arguments, while accepting them for mathematical argument is itself inconsistent.

 

[Jayhawker Soule]

We deduce by discovering a proof.

Perhaps. You would know best.

But just maybe we validate by discovering a proof to that which ws deduced.​

 

[Shadow Wolf]

Mathematics is a language we invented to describe our discoveries of the nature and functionings of the natural world.

 

[vulcanlogician]

No, it says something about right triangles *if Euclidean geometry is assumed*.

Understood. And I concede all my other points. You've answered those objections very well, especially concerning gravity. And you've also shown that the Pythagorean theorem is not true in any kind of universal sense.

But I was wondering: are there axioms that are shared between both Euclidean and non-Euclidean geometry? (I honestly have no idea.)

If there aren't, then you can go ahead and ignore the rest of this post.

But if there ARE, then could a person (using JUST those axioms) form the "nub" of a (very limited) mathematical system? I'm being very hypothetical here. I'm not saying this geometric system would be very useful for anything... my question is: could such a system exist? And if it DID exist, would IT be discovered or invented. Hopefully that question makes sense. I admit, it's a weird question. Maybe even a silly question now that I read back over it.

I'm working under the assumptions that the axioms themselves are absolutely discovered, ie. not invented at all. After all, if someone thinks the underlying axioms are invented, then of course all conclusions that follow would also be invented.

Or maybe you could take it the other way and say that the non-Euclidean and Euclidean axioms all sort of "count." Euclidian talks about flat space. Non-Euclidean talks about curves. But, once one figures out if the space in question is flat or curved, the theorems that deal with that particular space are objectively true.

Your thoughts?

 

[Heyo]

Understood. And I concede all my other points. You've answered those objections very well, especially concerning gravity. And you've also shown that the Pythagorean theorem is not true in any kind of universal sense.

But I was wondering: are there axioms that are shared between both Euclidean and non-Euclidean geometry? (I honestly have no idea.)

All but the fifths. Non-Euclidean geometry is the same with just the parallel Axiom dropped or modified.

 

[Shadow Wolf]

When?

Something like Pi, that the circumference of a circle over the diameter will equal something approximately related to 3 was first discovered back in very ancient times (like Egypt and Mesopotamia ancient) and would be realized again and again by multiple cultures that advanced far enough for the luxury of allowing people to focus on work this stuff out.

 

[Heyo]

I think @Heyo was getting at the fact that it is common in math for the same idea to appear independently in different places and even at different times.

Exactly. The basics of maths were found independently in Mesopotamia, India, China, South- and Mesoamerica at different times.

One example is the development of calculus, which is attributed to both Newton and Leibnitz (although others did contribute). Both came up with essentially the same techniques independently.

That is an advanced example.

The 'where' of discovery seems to be of less interest, though, than the 'who' and 'when'. Of course, different concepts in math arose at different times. Even today it is common for two researchers to find essentially the same results and techniques even though they have no contact.

That does suggest 'discovery'.

I think it, at least, hints at it. When something like the basics of maths are discovered multiple time - with no counter example - it is a strong hint that that is the only way to basic maths.
Compare that to religion, the believers don't agree on the number or the attributes of the gods. We can pretty clearly assume gods to be invented, at least more so than maths.

 

[Heyo]

Even math cannot be proved to be consistent (once it is strong enough to model the natural numbers).

Wait ... as far as I know, except from very esoteric fields of maths, it is consistent, i.e. free of contradictions. It is incomplete, though, i.e. not every law can be proven or disproven.
I'm thinking about the Barber's Problem or the Tarsky Paradox. Those are examples of incompleteness, not of inconsistency.

 

[ajay0]

I would say mathematics at its basic level incorporating zero was discovered rather than invented. The roman numeral system did not have zero in its numerals and western philosophies rejected the concept of emptiness or nothingness.

The modern numeral system including zero used around the world originated in India, and has its background in Indian religious philosophies focused on enlightenment.

The religious philosophies of Sankhya formulated by sage Kapila as well as the Shunyata philosophy of Buddha provided the philosophical conceptions of the numeral system and zero respectively.

Sankhya means number in Sanskrit, while Shunyata meaning void or emptiness provided the numerical term 'Shuny' meaning null in Sanskrit, which became 'sifr' in Arabic and 'zero' in western languages, when Indian numeral system and mathematics were adopted by the Arabs and Europeans later on.

The Indian numeral system and arithmetic reached Europe in the tenth century A.D, but was generally rejected on account of its foreign origin.

It was found to be superior to the roman numeral system in terms of accounting and indepth calculations, and after initial resistance in Europe for a few centuries, was finally accepted by western scholars as a valid and superior mathematical system for calculation purposes around the thirteenth century AD.

Hindu-Arabic Numerals: History & How They Revolutionised Mathematics | HistoryExtra

The evolution of Hindu-Arabic numerals revolutionised mathematics, astronomy, engineering and science which has been the foundation of much of modern life

www.historyextra.com

It is impossible today to imagine a world without what are known as Hindu-Arabic numerals. These nine symbols or digits (1 to 9) and zero (0) – crucially, together with the system of arranging them in place value – are at the very heart of so much of modern life. Without them there would be no computers and no space travel. Indeed, there would be precious little science, technology or medicine of any kind, to say nothing of mathematics itself.

Try to add LIX to VII and you’ll quickly understand why maths would not have evolved far with the Roman numeric system.

~ Violet Moller (historian, writer)

 

[Brickjectivity]

I can see the problem of deciding.

 

[sayak83]

Even math cannot be proved to be consistent (once it is strong enough to model the natural numbers).

I would strongly object that the axioms of math are discovered. We might observe what happens with very small numbers (say, those less than a million) and use that as an intuitive basis for forming an axiom system. But that is *us* making up the axiom system to fit our intuitions, not something that is forced by our observations.

For example, there are axiom systems that allow every specific arithmetic statement (like 341+782=1223) to be proved, but where commutativity of addition cannot be proved. Such a minimal system is completely consistent with our observations, but even simply general results can fail.

I would like you to comment on the Structuralist ontology of math if possible.
Structuralism, Mathematical | Internet Encyclopedia of Philosophy

 

[wellwisher]

I go with a mixture of discovery and invention.

Usually, mathematics starts with a vague intuition. We then invent formal systems that capture some aspects of that intuition. After that, we discover the consequences of the rules we invented.

Independent discovery generally happens because there are similar intuitions flying around. Often the specifics are different as invented by different people. Over time, the community decides which concepts are most productive as mathematics and those become standards assumptions.

I don’t think that mathematical objects are “out there” in any meaningful sense. We choose what is of interest and study those things. We choose our axioms to show that study. Unlike physics, say, I do not think an alien race would necessarily arrive at the same results in point set topology as we have.

The idea of independent discovery was correlated by the late Psychologist Carl Jung, with his theory of the collective unconscious. The collective unconscious would be analogous to brain firmware, behind human consciousness, that is common to all humans by virtue of our shared human DNA; genetically neural firmware evolving and defined over eons. These firmware or archetypes define us as a species; our collective human nature. Like the organs of the human body they are defined and refined over time on earlier foundations; archetypes.

Jung proved his thesis by showing many examples of very similar ideas appearing in many places with no proof there was any transmission of information. Our human mind's operating system are structured so similarly; eons of selective advantages, very similar conclusions or innovations would also be reached; consciousness will funnel to similar places. The steam engine was first invented in BC, but had little use until 18th century AD, where it appeared spontaneously, again. We have 6000 human language in earth, which would not needed or expected if there was only direct transmission, but no spontaneous innovation of similar types; language systems.

I tend to believe that math is a tool; fancy looking glass, that has many uses and has undergone refinement and lateral advancements; radio telescope. Mathematics is also like a faithful horse, that will go anywhere you lead it. We use math in computer games, to allow for endless lives. If you assume math is pure and natural, that means that this math not only predicts the after life but immediate reincarnation. I tend to believe that math is really more like a good horse, and will follow the lead of the human driver; will haul our foundation premises to market. All theories will use math to express them, even though most will eventually be replaced; different drivers of the same horse.

In the life sciences, statistical math is widely used. This is a very useful tool. However, the irrational bias, that math is pure and natural has led many to erroneously conclude, life is based on the horse and not the driver leading. It is quite bizarre, discouraging reason and common sense.

Division by a fraction is interesting in that allows us to violate conservation principles. If I have 1 kilowatt hour of electric energy and was to divide it by 0.5, I will get 2 kilowatts of energy. This violates energy conservation; energy cannot be created or destroyed, even if a most basic version of the math tool allows it on paper. This can occur when the driver has had too much to drink and the horse is leading. The faithful horse can know your routine and can even follow your unconscious lead.

In terms of life, life is very organized. The DNA is designed for specific template relationships. This allows life to replicate and duplicate itself with amazing accuracy over trillions of cells. So why use the statistical math horse, that is more geared to disorganized and unpredictable things; Lady Luck? It is like using a junkyard horse, who is used to pulling heavy scrap metal, to haul delicate produce. This will turn the perfectly ripe tomatoes and peaches, into bruised fruit, that appear less uniform at market. This is irrational for science; using the wrong math tool. If the horse is allowed to lead, the mind looks for flaws that are not there and then adds them; hammer leading the carpenter; we are all at risk.

My guess is this is about making money. There is more money in sickness and flaws than in genetic perfection. Although 1% may be sickness and flaws, this is 95% of the revenue, therefore everyone has to use the gold digger math and worship it as leading. The insurance company will also make money betting on accidents, which are a tiny fraction of all driving hours. We are taught to look at driving into terms of exception and that math horse, since this is where the money is lost and gained.

At least biology, now has a reason for its poor math horse choice. It should not take 10 years to come to market with a new drug that can save lives. This is due to the same math horsed used by Government Bureaucrats who think in terms of their career path and will stall until they are not liable or at risk. They do not think in terms of lives saved, but careers advanced against Lady Luck.