[Lewis Thomson (OP)]

Donald needs help with this existential question,

"Is there an unassailable excellent explanation why maths work? I'm not even clear on whether numbers are real, as in an objective thing or just a convenient useful imaginary conceptual tool. And why is there so much maths? One would think it would simplify, rather it gets more complex, diverse and unintuitive."

This is a colossal question, can you help with answering it? Leave your comments below...

[alancalverd]

You can start with numbers. The concepts of "more" and "less" are intuitive and important, and it seems that most animals can discriminate between objects and count, so addition and subtraction "work" as long as the objects are reasonably permanent and similar. This leads to basic arithmetic on the one hand, whilst the intuitive matters of "similar and permanent" are formalised and generalised by group and set theory when we appreciate the difference between "all" and "some".

Civilisation is specialisation. Once Ug started making arrowheads for Og, Ig and Eg, he probably set out his wares in 2-dimensional sets and got some idea of multiplication to work out how many he needed to make if all three hunters wanted two handfuls of arrows. Division is a bit more subtle, but if he puts all his arrowheads in a bag, it's useful to know how many to give each customer so they all have the same number with not more than two left over.

The moment you employ different craftsmen to build something, you need to formalise your concepts of shape and size, hence geometry. Whilst everyone is (or should be) astonished at the logistics involved in transporting and erecting the stones at Stonehenge, what amazes me most is the lintels that join the tops of the inner ring. I can imagine bashing the outer lintels until they fit (though I don't know what I'd bash them with) but the inner ones apparently sit on pegs, so you need some agreed form of measurement to fabricate the pegs and slots to the required dimensions, 6000 years before Ikea could do it with wood. And it seems that most of the stones arrived on site pre-cut and dressed  to size - virtually unheard-of (apart from bricks) in the building trade until the 1940s. Somebody, somehow, taught the Wiltshire  architects and the Welsh quarry masons the same system of measurement.

What has changed during recorded history is that maths has gradually overtaken the demands of trade and engineering, and provided insights into hypothetical universes that allow us to model bits of physics we haven't yet discovered.

The key to all this is the essential selfconsistency of mathematics. Every proof begins with an "if" and proceeds with the most rigorous tests towards a conclusion  that is absolutely and universally true if the initial axioms and "givens" are true. Thus we can build huge and completely robust mathematical structures on foundations that may be valid in certain circumstances, and may even turn out to be useful.

Apart from prime numbers, I can't think of any aspect of mathematics that isn't intuitive, at least to the extent of formalising an intuition so that two people can share and discuss it. But then I'm an oily-fingered experimentalist! 

[Eternal Student]

Hi.

Is there an unassailable excellent explanation why maths work?

This is the standard explanation:
    Axioms have been established.   The rest of what we would call "Maths" is based on using these axioms in a logically consistent way.   

This is the general discussion of axioms that goes with it:
   There is nothing and no way to prove the axioms are "correct"  or represent something that is actually true in nature.   Indeed the axioms do not even have to describe any type of thing that could be identified as something that exists in nature. 
     Just to be clear then, there isn't just one system of mathematics.  There are many systems of mathematics, although at school you would have focused on one system which was based on the axiom system known as  ZFC  or   ZF and I'll often write ZF(C) because the difference between the two isn't worth worrying about here. 

Here's a pouplar view:
    This system is NOT guaranteed to be a perfect reflection of things that exist in nature, it is just a logical development, a body of knowledge, based on some fundamental axioms.  Also, it doesn't matter if those axioms are  fundamental truths in nature or just abstract.  Mathematics is that which will follows from those axioms by the application of formal mathematical logic.  There is no way to measure the truth of any mathematical statement except by applying the same formal mathematical logic which was used to construct and develop the Mathematics.   So the validity of mathematics as an abstract object is "unassailable" as you phrased it.
    It's just a bonus that this ZF(C) system of mathematics does also seem to reflect things that happen in nature.  This is aided by the careful choice of axioms on which we now consider it to be based - which actually brings us to the next important point.

The real development of mathematics
     It is often portrayed as if the axioms of mathematics were developed first and modern mathematics just follows from this.   However, this is not at all how Mathematics developed in History.   A body of knowledge and techniques that we would call "Mathematics" was being developed and used before the modern axiom system was developed.  So, it's not just good luck or coincidence that Mathematics follows from the axioms:   The axiom system and the entire concept of formal mathematical logic was developed and proposed as the underlying mechanism of formal Mathematics after we already had some knowledge of Mathematics.   So, the modern axioms were hand-picked and the formal logic system was proposed, modified and polished to make everything work.  It is only with hind-sight that we can now look at mathematics and determine or decide that these axioms and formal logic are the most fundamental or basic building blocks of mathematics.
    Indeed this view is still considered to be a little artificial by many mathematicians.  We can show that all of mathematics can be created from these building blocks, so you can consider sets and axioms of set theory as the fundamental blocks if you want.  However, there might also be some other type of basic building block(s) that would be just as good as an underlying description of what Mathematics is.
   
Recent developments
   There have been some interesting results in Mathematics such as Godel's incompleteness theorem.  We once thought that we could prove any true result in Mathematics from the axioms but in fact we can't.  There are some true statements in mathematics than cannot be proven from the axioms.  So, if you regard Mathematics as a body of knowledge or true statements, then you could argue that mathematics is actually MORE than just that which follows logically from the axioms.

Summary:
     We can argue that mathematics is an abstract thing which is based on axioms and it is self consistent, so that it works and its validity is "unassailable" as you phrased it.
     However, there is no guarantee that this system is more than just an abstract thing.  In particular, the connection between the axioms and what happens in nature is uncertain.
     Over the many years that human beings have used mathematics, we have built a system that does seem to be useful.   If this was a piece of "science" we would say there is a lot of evidence to support the validity of mathematics.

I'm not even clear on whether numbers are real,

   You and many others.  It is sufficient that they exist as mathematical objects in an abstract system.  There is no obligation for numbers to exist as anything tangible in the real world.

   

And why is there so much maths? One would think it would simplify, rather it gets more complex, diverse and unintuitive."

1.     Human beings have tried to understand their surroundings for many years.   So a lot of knowledge has been accumulated.   Mathematics did seem to be useful and successful, so this knowledge has been retained, valued and passed on.

2.     It really is possible to reduce a lot of Mathematics to a much smaller and simpler list of facts and a few techniques you can use.   This is what the axioms and the formal system of mathematical logic actually does.   However, human beings are not computers.   We do not have the processing speed to solve problems involving mathematics starting just from the fundamental axioms each time.   Neither do we have the insight to realise which axioms and techniques will eventually solve the problem and we certainly do not have the time and processing speed to just use every possible combination of axioms and techniques until the problem is eventually solved.   So it is better and more natural for us human beings to retain a larger set of facts (theorems and results about mathematics) and just build up from there.   For example, I can just retain the knowledge that the sum of the angles in a triangle is equal to 180 degrees and use this,  I do NOT need to prove that the sum of the angles in a triangle is 180 dgreees every time.
   
     Just for your interest:   There is research being done into developing computer systems that will act as a Mathematician and potentially replace the need for a human mathematician entirely.  You can google "Computer assisted thereom provers" which include ATP or (fully) Automated Theorem Provers if you want to know more.  The current systems frequently just use brute-force techniques to prove a theorem directly from many possible combinations and manipulations of the axioms.
    Anyway, that's the best answer I can offer for why there is so much Mathematics and so many results that do seem complicated or un-intuitive.  We are not computers and the shortest list of facts and techniques is not useful for us. Every theorem in Mathematics is a short-cut we can use, rather than proving everything from the axioms.  It also allows us human beings to work on problems in a way that is far more intuitive for us.   For example, it is easier for us to know that a piece of wood will just behave like a piece of wood and using this knowledge we can cut and glue the wood to make a picture frame.  It would take an incredibly long time to make a picture frame if we could only see a piece of wood as a collection of atoms and had to consider all the possible ways we can manipuate those atoms.

Best Wishes.

[evan_au]

I take it back to the psychology of "fairness".

Humans (and animals) have an innate counting sense called subitization. But it's not exact.
- Humans (and some animals) have an innate sense of fairness, and get really annoyed if they feel that they aren't being treated fairly
- Some of the earliest clear examples of mathematics is in accounting records on clay tablets. The traders wanted to be sure they were getting a fair deal (and the king wanted to be sure he got his "fair" share of the deal, too)
- Less clear examples of mathematics are in (what appears to be) tally marks on sticks. Some suggest this may have been keeping track of sheep herds.
- or even earlier, possibly tally marks on cave art. Some suggest this may have been keeping track of kills in the hunt.

Now we count different things - petabytes of data on the internet, exponential growth in COVID infections, chemical reaction rates inside a gas turbine, rates of species extinction or billions of light-years to a distant galaxy.
- But we still argue about fairness in the national budget, what is the value of a species or clean air, or in the distribution of COVID tests, what are the odds of the James Webb Space Telescope getting commissioned on schedule, or which kid gets the larger slice of cake...

https://en.wikipedia.org/wiki/Subitizing

[nicephotog]

If you think about it (i am), Math is the ability to subsection a part of the infinite expanse of existence to a finite piece as a set of parameters and bounds that is given actual cohesive reality of place time and space by being proportional and defined giving a small mind assurance of the finite piece' reality as actual.

[Eternal Student]

Hi.

I take it back to the psychology of "fairness....".

   That's interesting evan-au.  Obviously that's the pessimistic view of things.   Maths developed because human beings were very self-interested or selfish.

If you think about it (i am), Math is the ability to subsection a part of the infinite expanse of existence to a finite piece as a set of parameters and bounds that is given actual cohesive reality of place time and space by being proportional and defined giving a small mind assurance of the finite piece' reality as actual.

    Hi nicephotog and thanks.

Best Wishes.

[evan_au]

Maths developed because human beings were very self-interested or selfish.

I agree that this view of the origin of maths is rather cynical.

But the maths of human psychology has some weird characteristics:
- It is logarithmic in nature. Receiving $1000 perceived very differently by someone with $100, compared to someone with $1 million
- It is asymmetric in value: People value losing $1000 as much as gaining $2000 (on |average|)
- It is asymmetric in time: You tend to remember the end of something better than you remember the beginning

With these wild variations in value, and the fallibility of human memory and biases, it makes sense to have:
- a written form of numerical records,
- together with associated standardized weights and measures
- and a standardized currency (whether shells, gold or minted coins)
- And defined conversion rates between these things to fairly assess the value
- And this would lead to a system of arithmetic

Similarly, in Egypt, the annual Nile floods washed away the property markers
- So they needed written records of property ownership
- And accurate surveying techniques
- To fairly divide the farmland every year
- Which would lead to a system of geometry

We are familiar with the story of Archimedes' "Eureka moment"
- He discovered density and buoyancy, because...
- King Hiero of Syracuse thought that he had been cheated by his goldsmith 

The Indian decimal numbering system was enthusiastically embraced by the Arabic world because it simplified the task of fairly dividing an inheritance according to Moslem law.
- Apparently, this process required solving problems in algebra
- So maybe it is no surprise that algebra is named after a Moslem mathematician...

See: https://en.wikipedia.org/wiki/Prospect_theory

[Colin2B]

Maths developed because human beings were very self-interested or selfish.

I agree that this view of the origin of maths is rather cynical.

I don’t think it is cynical. Most early inventions are born of a demand or need. Later on people start to look at patterns and principles, but that demands a degree of leisure (thinking time) or specialisation (allowing experimentation) within a culture.
The evolution of technology and discovery is a fascinating area, full of dead ends and interesting chains of events.

[alancalverd]

You tend to remember the end of something better than you remember the beginning

Scarcely relevant but fun: I used to play in a band that only rehearsed introductions and endings - rarely more than 32 bars per piece, on the basis that these were the bits people remembered and judged. Underlined by Lesley Garrett (soprano) who commented in a radio show "That was me versus the London Philharmonic Orchestra. We started together and finished together and I can only apologise to Herr Mozart for the bit in the middle".  And whenever I've designed a clinic, I insist on really good presentation of the toilets and coffee machine: people remember the familiar bits and forget the x-ray room or operating theater.

More seriously, the psychology of maths is indeed very important when dealing with statistics.We do indeed have an inherently logarithmic response, which is why I have advocated a "risk index" for public communication:

If the probability of an undesirable event is P, the risk index R is 10 + log₁₀P.  Thus inevitable events have R = 10 and if R ≈ 1 no living person is likely to have experienced it. 

The practical application already exists. Most people are happy to accept R_death = 4 for activities in everyday life. Where R ≈ 5, we may undertake an activity (risk sports) with a significant reward. If R > 5 we look for legislation and licensing, and we tend to ban activities where R > 6.

The case fatality rate for COVID has R ≈ 8 at present. Seems like a good reason to wear a mask.

[evan_au]

Maths developed because human beings were very self-interested or selfish.

I gess we could expand this to say that many areas of science & technology arose because the king/emperor/despot/ruling council was self-interested and/or greedy.
- They didn't want some other king/emperor/despot/ruling council to kick them out and take their place
- And/or they were eying some other king/emperor/despot/ruling council and wanted to take over from them
- Early astronomy/astrology was promoted as the way to find the most propitious timing of these forays...

This led to the development of :
- metalworking for swords and spears (in bronze, with spinoffs to sculpture and iron with spinoffs to plowshares),
- military technology (the Romans brought this to a high level, with spinoffs in plumbing and building),
- We've mentioned in other threads military communications (with spinoffs in music and more recently, the internet)
- Galileo's telescope was first used to detect approaching ships, to see if they were friend or foe (and to make sure they paid their taxes)
- The British Admiralty funded development of accurate clocks for navigation; more recently the US military funded development of GPS
- and more recently, nuclear energy and the space race.

Even the "pure" scientist has to get his funding from somewhere...

They will beat their swords into plowshares and their spears into pruning hooks.

...but it has often gone the other way!

[nicephotog]

- The British Admiralty funded development of accurate clocks for navigation; more recently the US military funded development of GPS

A couple of things you missed there are from WW2.
1. The use of solar cells for sabotage by sending an electronic signal to the detonator control. When a train exited a tunnel the solar cell would build charge and trip the semi-conductor (probably coupled with thermionic valves, aka vacuum tubes) and trigger the explosive.
2. The naval guns had electronic (basically radar) range finding which is why the Captain of the *HMAS Sydney (*WW2) was posthumously prosecuted for approaching within 12 miles of the German raider vessel off the west Australian coast.

1.

Just a piece more interest...

[alancalverd]

- military technology (the Romans brought this to a high level, with spinoffs in plumbing and building),

The sister of an old friend (I never met the sister) wrote an interesting thesis ascribing Roman military dominance to standardisation. Hard-surface roads driven in straight lines made it easy to supply and reinforce  garrisons and settlements, but marching on roads demands boots. Rather than deploy bespoke shoemakers on the front line, they established factories that made and transported huge quantities of shoes in standard sizes so the infantry could replace them as needed.

Quite how they managed to survey and build their roads and bridges with arithmetic based on Roman numerals is still a wonder to me.