What is 'maths'?

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...

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! 

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.

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

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.

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.

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

Quote from: Eternal Student

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.

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.

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!

- 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...

- 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.

(alancalverd) Bees have an accountancy sense of more or less (heard on ABC Radio Australia science news), too when removing honey frames from the hive as a keeper, after a particular ratio is removed and displaced it is wise to wear beekeeping gear!
Bees do have a concept of quantity.
More or less is the general quantifier measurement value of "thuggery".
It is interesting to note in a computer you cannot physically store one or zero as "no charge" in the capacitance, only small charge (weak) or large charge (strong), it's unpractical because of component leakage, it better to take a specified measurement parameter to understand everything was operating correctly.
Have an optical sight gag of possible thuggery at 20m on the bar of gearth or maps.
19.235535,18.289995
Which is higher or lower surface?

I think mathematics is something we consider is a tool. For a long time the only counting we did, presumably, was because we traded goods.

But then mathematical truth is something that again presumably, has always been true. Pythagoras' theorem was true before it was written down. It was true before humans evolved. So when was it "first" true? Is that even a question?

If you want to study postgrad math, you need to get your head around constructing algebras, and proofs. Pretty much all the current research is about constructible mathematical theories. Physics likes to borrow the ideas and see if they have any practical use, in particle theories (physics joke there).

There are some pretty damn good lectures online. My understanding of what is an algebra is now, it's something you can construct. You can start with almost anything, a set of things is better though.

So an incidence algebra of a partially ordered set is given, you define all these operations and show that it gives you a Hopf algebra, or at least a bialgebra. It's really just an example of how mathematics is another form of art, all art is intellectual I think.

If anyone wants to know more, this example explains in very simple terms, what is a tensor product, and how the arithmetic operations of addition and multiplication can be defined on two dual spaces, even what is a homomorphism and why it's a good thing.

Hi.

But then mathematical truth is something that again presumably, has always been true.

    Well maybe.

    There are problems proving that any system of Mathematics (which must inlcude some basic arithmetic) is consistent.
   See    https://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_theorems#Second_incompleteness_theorem
 or more generally,  Godel's first and second incompleteness theorems.

    It may be consistent but we can't prove it.   This undermines the pedestal on which Mathematics is often put.   So the example you gave, Pythagoras' theorem, fortunately is something we can prove from the axioms and would seem to be true by any reasonable judgment.   However, it's also true and provably true in an axiomatic system where we have just deliberately added Pythagoras' Theorem as one of the axioms and arbitrarily knocked out some of the other axioms.   There doesn't seem to be an objective axiomatic system that we can have and also KNOW that this system will be consistent.   
      It's true that cheese can fly in a system where I say that it can, that's just not a very good system to use and any system that inlcudes that axiom could quickly become demonstrably inconsistent.   Mathematical truth is something you can give any statement if you just start with that statement as an axiom -  so it's worth nothing. 
      What we would want to use is a system that is consistent.  So what you probably mean by "mathematical truth" is that the statement is true under the systems of mathematics that we do commonly use.   We hope that our preferred systems of Mathematics have intrinsic validity and integrity such as "consistency" - but we will never know.
      I do not know if Pythagoras' theorem is true now, was true or has always been true in any absolute sense.   I only know that it is true now and would always have been true in the axiom systems that we do commonly use, for whatever good that axiom system may be.

Best Wishes.

So what you probably mean by "mathematical truth" is that the statement is true under the systems of mathematics that we do commonly use.   We hope that our preferred systems of Mathematics have intrinsic validity and integrity such as "consistency" - but we will never know.

Except that certain mathematical truths do appear to be entirely consistent with modern physics. So it's always been true that, say, in any given quantum state time is always linear. This is something that is part of proving that a given sequence of quantum "gating operations" on pairs of or on single states will halt properly, with an output.

So quantum logic gives us a kind of backwards narrative: to prove that a theory is mathematically true means proving it can be an algorithm with no errors. No logical errors. If a program works correctly it will have always worked correctly even before anyone builds a computer.

But yes, Godels theorem and Turing aside; we can still prove things except we need axioms and we can't do more than assume they are true. Then we use the logic based on that assumption to show there are no contradictions (or errors in the algorithm).

But, yeah,  time is linear. Who knows why that's true?

I'll try to expound a little: if the only way to "prove" that time is linear in quantum experiments, up to measurement, then what is measurement? How do you prove you measured the output of some internal state? It seems you can't do this algorithmically without assuming that measurement is possible even if the mathematical formulation (in the theory) doesn't predict any such thing.

Moreover, Godel's and Turing's theorems have the same problems with axioms. Time is axiomatic, since, we can't keep any around to prove it exists.

why is there so much maths?

Because different maths is required for different situations.
- A child may say that the number after 12 is 13 (when they are talking about the natural numbers)
- Except when they are talking about a clock, in which case the  number after 12 is 1 (a modified form of modulo arithmetic)
- Both are valid answers, in their own domain

There is a debate about whether maths is invented or discovered.
- Some ancient Greek mathematicians believed that geometry held an objective reality in nature, and humans just discovered it

Hi.

But, yeah,  time is linear.

    I'm sorry, I'm not sure what that means.   Time varies linearly with the number of seconds that have passed rather than having some quadratic dependence?   Time is an operator that is linear - time acts on one girl to make an old lady,  so it acts on  λ girls to make λ old ladies?   I suppose all those things are true of time.

    A quick perusual of the web suggests people say "time is linear" when they mean one or both of the following:
1.  It's uni-directional.  You can't go backwards.
2.  It has some beginning and some end with all things in between being conceptualised as a line rather than something else like a complicated branching structure or a fuzzy cloud of stuff.   So all events are totally ordered.  Specifically given any two events we could always say one event is "before" the other.

then what is measurement?

   An excellent question but not one with a short answer.   It probably needs its own forum thread.
It's a contentious issue in Quantum mechanics which I'm sure you're aware of.

Best Wishes.

time is linear.

There is a debate about whether time is continuous or quantized; unfortunately, at this point in time, we don't have much evidence either way.
- At the scale of General Relativity, time is treated as continuous
- When you get down to the detailed behavior of quantum fields in the vicinity of a black hole event horizon, you end up with some stubborn infinities, which have so far resisted attempts to unify General Relativity and Quantum Field Theory.
- One possible interpretation is that time is actually quantized (at some level).
- In support of this, some researchers found that when they used a computer to simulate cases where these singularities appear in the theory, they don't see them in the computer. One aspect of computer simulation is that you have to quantize the simulation in both time and space
- and/or maybe space is quantized; as Einstein showed, Time and Space are both aspects of a unified Spacetime.