What is “Mathematics”

[J HancockParticipant @jhancock95746]

Let's be honest, we are only where we are today because our early relatives did things 'empirically ' , without

the precise knowledge of 'mathematics ' to predict what would happen if they built them that way.

Sometimes they went wrong , Tay Bridge , Tacoma , Comet 1, etc.

[Anonymous]

The question of what is mathematics has stumped mathematicians for centuries, but particularly in the 19th and early 20th centuries. Questions like what makes a mathematical system useful, and what are it's characteristics? Or how many infinities are there and what are their properties. In the 19th century it became clear that there are limits to mathematics, but that there are also alternatives to what most people considered to be mathematics. For instance in this thread many people have referred to trig and geometry. What they actually mean is Euclidian geometry, ie, based on planes. There are a myriad of other geometries constructed on other surfaces. Another useful geometry is spherical geometry (a special case of a Riemannian geometry) which, being based on a spherw, has uses in navigation. Each geometry has different characteristids. In Euclidian geometry parallel lines never meet, in spherical geometry they always meet at a point.

In the 1920s Hilbert proposed a study to show that all mathematics follows from a correctly chosen set of axioms, and that said set can be shown to be consistent. This turned out to be impossible on both counts. In the late 1920s Godel showed that within any mathematical system sufficiently powerful enough to contain natural numbers there are true propositions about natural numbers that cannot be proved, or disproved, from the axoims. In simple terms any useful mathematical system is incomplete and inconsistent as a consequence of it's existence. I'm afraid that those posters who think that mathematics is always right, and consistent, are wrong.

There is nothing in mathematics that bears any relationship to the the real world, so I think PatJ is also wrong in his definition. Of course mathematics is very useful to model the world, but it is only a model, which may, or may not, be a good fit. But there is nothing inherent in mathematics that says it reflects the world. Newtonian mechanics provided a good model for describing the physical world up until the late 19th century, when new experiments and measurements seemed to be inconsistent with the Newtonian model. In the end that led to the development of relativity which extends the validity of the model, but has it's own inconsistencies.

Andrew

[Nigel Graham 2Participant @nigelgraham2]

Perhaps how we define Mathematics depends much on our own experiences.

Maths was my worst subject at school, and I disliked it intensely.

I am not good at abstract concepts, and sometimes need how things work to understand how to use them; while externally the subject tended to be taught as abstractions with little mutual reference, let alone with real things. Why did know where the cyclist from A to B will pass the walker from B to A?

Most school text-books do not explain anything clearly: a definition or two, a couple of worked examples and a set of exercises. They were written to support a teacher who could actually teach – as some of mine could not. Oh yes, they knew mathematics, but not teaching!

So that was my early dreams of being a scientist of engineer scuppered. (Dad was a Chartered Electrical Engineer working as an MoD Scientist; my uncle worked in British Railways' Technical Centre in Derby, though I don't know in what capacity.)

It was really my hobbies and later work that made me see Maths as a set of useful tools for real things.

Model-engineering uses much mensuration and some trigonometry; as well as arithmetic of course.

Geology made me see what Differentiation is. In a geology-club lecture on analysing river gradients to reveal rock boundaries hidden by valley sediments and vegetation, something made me write a simple formula in differential notation. Suddenly I twigged what differentiation does: show how much you go up (or down if you are a stream), for a give distance along.

Logarithms and the slide-rule were still in use at school, with calculators only just appearing in my last Year. I could use logs for times and power sums but had no idea how they did it. Until some decades later, as a lab assistant in a sonar lab, I became used to such glories as " -211.5dB re 1V/µPa " . db? Decibels? They be logarithmic, they be; mercifully all to base-10.

I found my way to understand them, by remembering first doing sums by writing "H T U" at the heads of the columns, to understand that any ordinary number is really a string of counts of integer powers of 10.

.

I have found elsewhere and typified above by Pat's accounts, an apparent much bigger difference in education across the Atlantic than simply maths / math.

The British school system has always given a selection of different topics within a maths syllabus itself part of the school curriculum. Hence the overall nature of the exams at the end of the entire course, even if over two or three papers to cover the syllabus. So you leave school with a qualification in Mathematics as far as the syllabus has taken it.

The American schools – and I invite PatJ to correct me if I am wrong – seem to treat each "math" topic as a completely separate curriculum subject. So presumably you might leave school with a qualification in geometry, and a qualification in Algebra, etc.

.

Many who struggled with Hard Sums consider Algebra as one of the sticking-points; perhaps because like me they found the abstraction hard. It has none of the obvious belay-points I later found in eccentrics' angles of advance, Dorset's rivers and measuring sound. (That strange value I quoted is the receiving-sensitivity I still recall, of a particular, commercial, laboratory reference-hydrophone.)

So I gave the matter some thought and realised that by starting as seeing it as shorthand for how to work out real-life things like areas, thence a shorthand way of writing the rules of arithmetic itself, it starts to make much more sense.

Going on from those, I became curious about the bats whose underground roosts I visit sometimes, as a caver. (We are very, very careful not to disturb these wonderful little animals.) In particular, what they might image acoustically when flying by sonar in total darkness: do their brains create something akin to the sea-bed images from side-scan sonar? I realised I needed understand not only deciBels (rather than just measuring them at work) but also how hearing works; using the human ear as a reference as finding the information for human biology is easier than for other animals. It also gives a neat comparison.

It led to one of the most staggering facts in biology I have met. The 0dB level for airborne acoustics is set by the minimum Sound Pressure Level in air for the fully-healthy human air; and it equals 20µPa. (Marine acoustics uses 1µPa . We can't use 0 as the minimum because that puts a divide-by-0 impossibility in the conversions.)

Let's explore it:

We need typically 6 Bar to propel a miniature steam locomotive.

Perhaps 3 to 4 Bar in our car tyres.

1 Bar is Standard Atmospheric Pressure – as we breathe at sea level.

1 Bar is also 100 000 Pascals.

So a Pascal is not much use in ordinary life… nor in sound. It's far too big for that! So we divide it by a million.

So 1µPa = 1 / 100 000 000 000 Bar.

I.e. Our 20µPa faintest sound pressure level at full otical health, is just a five-thousand-millionth of atmospheric pressure…..

Puts the whispered sweet nothings barely audible in the darkness of the boudoir, in a rather different light!

[SillyOldDufferModerator @sillyoldduffer]

Posted by J Hancock on 29/05/2022 11:37:02:

Let's be honest, we are only where we are today because our early relatives did things 'empirically ' , without

the precise knowledge of 'mathematics ' to predict what would happen if they built them that way.

Sometimes they went wrong , Tay Bridge , Tacoma , Comet 1, etc.

The Tay Bridge enquiry found that although Sir Thomas Bouch had consulted the Royal Astronomer about wind-pressure, he hadn't kept up to date with developments on the continent or in the USA. If he'd read American sources and done their maths, he'd have realised his design wasn't strong enough. Consulting the Royal Astronomer, wasn't as daft as it sounds. At the time the Royal Astronomer was Chief Government Scientist, and done by a first-rate mathematician. He wasn't an engineer though, and not expected to read engineering journals.

I've always been suspicious of the Comet 'no-one knew about metal fatigue' excuse. Metal fatigue was a well known phenomenon when Comet 1 was designed, famous because it caused serious problems with WW2 Liberty Ships. One of them broke in half whilst tied up alongside a jetty! The main cause was found to be metal fatigue due to stress raising sharp corners on hatch covers and other openings.

I find it odd that aeronautical engineers designed square windows for an aircraft without realising the similarity with a maritime problem identified in 1943! Possibly they tried, but lacked the resources needed to do the huge number of calculations required. I think it possible that someone senior in the company decided to save money, or couldn't raise it, and then had to dodge responsibility. (Before electronic computers, many engineering calculations were replaced by approximations because they needed large teams of expensive people and took forever to do.)

It's interesting that George Boole cracked binary mathematics in 1854, almost a century before technology had advanced enough to build a digital computer!

Dave

 

Edited By SillyOldDuffer on 29/05/2022 13:47:07

[Martin KyteParticipant @martinkyte99762]

Posted by Andrew Johnston on 29/05/2022 11:49:57:There is nothing in mathematics that bears any relationship to the the real world,

Andrew

Thats an interesting statement. Maybe I could invite you to comment on the proposal that anything that can be proven to be mathematically impossible cannot exist as a real world phenomena which if true would demonstrate a connection if not a relationship.

regards Martin

Edited By Martin Kyte on 29/05/2022 15:10:32

[not done it yetParticipant @notdoneityet]

1854, almost a century before technology had advanced enough to build a digital computer!

Wasn’t the abacus a sort of digital computer?🙂. S’pose it still is, actually?

[Clive SteerParticipant @clivesteer55943]

I'm not sure I understand Andrew's comment that mathematics may not be correct or consistent. As far as I understand, and I'm not a mathematician, mathematics is a rules based system and we have defined the rules so how can it be inconsistent. Similarly computer instructions are a set of rules if the program you've written using them doesn't do what you need the fault is with the coder not the code.

However I do agree that the "mathematical model" may not reflect reality exactly and if assumptions are made or important factors are not included the model may not even be close. What we don't know or make assumptions about may be inconsequential or disastrous. In the case of the Comet 1, metal fatigue was known about , but for whatever reason was not considered to be an issue or maybe the cost of testing for it was considered prohibitive. However after the event the testing was done.

When explaining to young engineers the need for extensive product testing I would draw a circle on a piece of paper and say that what is inside the circle is what you want the product to do but what is outside the circle is what you don't want it to do. Mathematics may be used to substantiate what the product should do but may not be so good a tool to find out what it shouldn't do.

CS

[derek hall 1Participant @derekhall1]

Regarding the last sentence in Nigel Graham 2 wonderful post….

Only in this forum can we make the leap from "what is mathematics?" to "whispering sweet nothings in a boudoir" !

Brilliant !

[CalumParticipant @calumgalleitch87969]

Posted by Clive Steer on 29/05/2022 15:37:09:

I'm not sure I understand Andrew's comment that mathematics may not be correct or consistent. As far as I understand, and I'm not a mathematician, mathematics is a rules based system and we have defined the rules so how can it be inconsistent. Similarly computer instructions are a set of rules if the program you've written using them doesn't do what you need the fault is with the coder not the code.

The point is that when we do mathematics, we set some starting rules, called axioms, and then we deduce what those axioms imply. If we deduce something that makes no sense, we go back to our axioms and tweak them. The "rules" are implied entirely by the axioms and the use we put them to.

This has happened several times – Newton's laws have already been mentioned, but in fact Newton's entire edifice of calculus was in the early 19th century found to be built on weak foundations, and the modern concept of real analysis was born, one of the first "real" bits of mathematics any undergraduate does ('real' in the sense of 'real' numbers). This work gave rise to questions about what mathematicians could compute and whether the subject was open ended or whether it could be completely catalogued, work which turned out to be closely related to mechanical computation, and we all know where that ended up.

What Godel found was that once you have set up some axioms and deduced their consequences, there are true things that you can't prove with those axioms, and that you can't prove that the axioms don't result in something clearly wrong. In other words, can you start out with the axioms, do correct operations, and end up with a result like 1=0? Godel proved that you can't prove it. Similarly, any open conjecture, like the 3n+1 problem, might just turn out to be an example of a property of numbers that we can't prove using our normal mathematics.

If this feels unsettling, it should.

[PatJParticipant @patj87806]

Definitely some interesting discussions here.

I don't know enough about mathematics to make any sort of definitive arguement about what it is or is not.

The best I can say is that I use math to make predictions about how a design may work, and if my mathematical model is sufficiently correct, then I can predict how my design will work in the real world.

Perhaps a more accurate statement would be "The math that I use on a routine basis, on the planet earth…..", and not necessarily all math that is known to man.

Who knows exactly what happens in the center of a huge black hole; our math may be radically different under those conditions.

One thing I am grateful for are the Law of Physics, because under the conditions we live in, the Laws of Physics seem to always hold.

I had one client insist there was something wrong with the wires that were installed as a part of my design.

His HVAC units keep burning up. It got to the point where he was threatening to sue me for a bad design.

I told him that the utility company was spiking his lines, but he did not believe me.

I told him to go ahead and sue me, but good luck with that, because nobody had broken the laws of physics yet, and if he was the first, I wanted to buy all the stock in his company, and we would be rich beyond comprehension.

He finally got the utility company out with recording instruments, and low and behold, the utility company was spiking his service voltage, just as I told him.

He called me back, and he was astonished that anyone could predict such a thing without even traveling to the site to check things out.

Another Owner called me up and said my electrical system design was bad, and he had absolute prooof.

I said "How so ?".

He said "My electrician disconnected one phase of a live 3-phase circuit, and measured 69 volts across the open conductor".

I laughed out loud, and said "Congratulations, you have discovered the property of induction, upon which all 3-phase alternating power systems operate". I told him to go try and do something productive in life and give up the voodoo electrical stuff.

I went to his facility, and said "Show me the device you are having problems with".

We walked over to a large PLC controls cabinet, and he said "The components in that cabinet keep dropping out", ie: relays and such.

I looked around at the nearby PLC cabinets, all of which had a 120VAC input control power regulator.

This panel had no regulator. I said "Where is your regulator?".

He said "I don't know".

I said "Put a voltage regulator on this panel".

His problem was solved.

.

Edited By PatJ on 29/05/2022 20:01:37

[PatJParticipant @patj87806]

My dad and his friend use to make observations about things, and exclaim that "that proves that math is wrong", "or that proves that scientists don't know what they are doing".

One example was that a bumble bee was calculated to not be capable of flying, and yet it did, and so that proved that bumble bees defied the law of physics.

It never occurred to him that perhaps the forumula that they came up with for the flight of a bumble bee could be flawed.

A mathematical model is only as good as the person who came up with it, and if one is lucky, a formula may be accurate enough to allow predictions about how things will act in the limited range of things we experience on earth.

It surprises me how many people think that a mathematical formula must be infallable.

I recall one hilarious "Far Side" comic strip, where there were two scientists on a beach, with chalkboards.

One scientist had a small chalk board, and a few simple equations on it, and the other had a much larger chalkboard, with very complex equations on it.

The scientist with the larger chalkboard was surrounded by numerous bikini babes, who where fawning over him.

The other scientist sad alone on the beach.

Such is life and beaches and mathematics I guess.

.

 

Edited By PatJ on 29/05/2022 19:52:56

[OldironParticipant @oldiron]

Posted by derek hall 1 on 28/05/2022 21:09:10:

Ok so someone please clear this up for me.

Mathematics has an "s" at the end

I always shorten it to maths……which is logical, like the subject !

So why do some call it "math"……it cannot be short for mathematic……….?

One of lifes conundrums I suppose……

I hated maths at skool, I was hopeless. Managed to get an apprenticeship and suddenly with the right lecturers I began to grasp maths. Just goes to show the importance of a good lecturer or teacher…

Regards to all

I agree Derek, americanisms creeping into our language every where. I was taught maths at school. My FIL is a Professor of mathmatics and a Physicist he has taught all over the world. . He says "maths" is what he was taught at Uni and maths it is. So who am I to argue with him.

regards

Edited By Oldiron on 29/05/2022 20:00:33

[PatJParticipant @patj87806]

I am amused at many things that the modern videos reveal to us about very similar but sometimes different cultures/countries.

I recall hearing the first casting video from someone in the UK, and he said "We will use AL-U-MINI-UM".

I busted out laughing. I said "He must mean ALUM-IN-UM.

There are many words that strike me as amusing.

Bin, lorry, bonnet, nappy, etc.

Even in the states, you don't have to go very far to find language differences.

The yankees have their yankee-speak, as do the southerners.

And the Japanese language is really amusing to me.

Rear view mirror is something like "Looky-backy".

Front windshield is roughly "frontagrassa".

If feel very fortunate that the language differences are minor between the US and UK, but of course many of our ancestors in the US did come from the UK, and so we are basically transplanted UKians (I created a new word) .

I tried to learn Danish one time. I had no luck at all with that.

German I think I could learn without too much difficulty, and I could probably pick up French without too much trouble. I can speak some Spanish, and had two years of it in school, because I refused to take Latin.

But Danish, forget it, you can have that convoluted mess.

I guess one slant on math(s) is that it is the universal common world language.

.

Edited By PatJ on 29/05/2022 20:14:58

[PatJParticipant @patj87806]

I had a cousin from Detroit (I live in the South US), and she would visit.

She would say "Do you have a pop?".

I would say "Yes, but Dad is at work now".

She would say "No, do you have a soda?".

I would say "Yes we have plenty of baking soda in the pantry".

She finally would say that she wanted a canned drink.

I said "Oh you want a Coke", (as in Coca Cola).

"No I want a Sprite" she said.

I said "Well Sprite is a type of Coke in these parts".

LOL, this would go on for hours.

When I visited Detroit, I would ask for a bowl of grits (hominy) at restaurants.

They looked at me like some sort of space alien. Nobody up north seems to have any clue about what grits are, and they sure as heck don't eat grits (they consider that horse food).

.

Edited By PatJ on 29/05/2022 20:24:12

[Anonymous]

Posted by Clive Steer on 29/05/2022 15:37:09:

…mathematics is a rules based system and we have defined the rules so how can it be inconsistent.

Calum has explained why that is incorrect. I would add the following to the explanation. Mathematics is not a rule based system, it is an axiomatic system. That is one starts with a list of axioms and then one can create rules that can be proved by reduction to the basic axioms. An axiom is a statement that is taken to be true but is unprovable.

It is possible to create a mathematical system that is complete and consistent. What Godel demonstrated is that a mathematical system that has axioms that allow the existence of natural numbers (in other words is useful) to be derived is incomplete and inconsistent.

The same is true of computer programs; they are not deterministic. Much effort has been expended on developing axiomatic bases and formal verification for computer languages to try and ensure that they will do what is intended. I believe it was Tony Hoare who stated that within any large program is a small program trying to get out. As a starter he invented Hoare logic in an attempt to prove program correctness.

One is highly unlikely to come across inconsistencies in mathematics that excite pure mathematicians. The engineer can safely assume that the mathematics we use will work.

Andrew

[Anonymous]

Posted by Martin Kyte on 29/05/2022 15:09:07:

Thats an interesting statement. Maybe I could invite you to comment on the proposal that…..

I obviously didn't explain the idea very well. What I meant was related to the statement by PatJ that mathematics is used to describe the universe. Of course it is, but the point I was trying to make is that mathematics does not exist for that reason. It exists in it's own right and is independent of any practical application, at least to pure mathematicians. Mathematics is a useful tool, but is not determined by physical applications.

Andrew

[PatJParticipant @patj87806]

I think many will have their own ideas about what mathematics is or is not, and I am pretty sure there may not be a convergence of all the opinions.

Perhaps a more accurate title for this topic is "What is mathematics to me, and how do I use it to make a living?".

Some facinating discussions for sure. I am humbled by the depth of thought.

.

[Nigel Graham 2Participant @nigelgraham2]

Sometimes I think rather waspishly that a "pure mathematician" is one for whom the best mathematics has no physical, let alone practical, application at all; but are there in fact any branches of maths like that?

Does mathematics at very advanced level diverge from physical things that work to numerically-consistent ways, to areas that work in themselves if you plug simple numbers like 2 and 3 into them, but which describe nothing real at all? Becoming really just very hard number-puzzles that never find problems to solve other than themselves?

I don't include rather pointless stunts like calculating pi to the umptee-umpteenth decimal place (I think that's an infinite trigonometry series, but well above my pension grade), but rather arcanities like the theories of numbers themselves.

For it reminds me of an incident at work…

One of my superiors asked if he could use my computer, in a small room off the laboratory, to find a reference to a paper concerned with the particular engineering-physics experiment he was carrying out. It saved him having to traipse back a very long way to his office.

When I returned to it I discovered he'd become side-tracked (he told me when I asked), and had opened a paper in English, by two Russian mathematicians. I say "English". There were not many words, but they did reveal the Muscovites were offering an alternative way to solve some equation bearing what looked like a French name as its … inventor? discoverer?

I have no idea what it was about, in fact I had never even seen many of the symbols in about three screen-fulls of pure algebra. It might find some very arcane application, perhaps in quantum physics or the like; but the paper was pure and unsullied by any such references to trade.

'
I rescued from a works clear-out a coffee-mug apparently from a museum gift-shop. Its artwork commemorates the 18C English mathematician George Green, and bears the following quote from his work. It makes sense if you read all four question-marks as Integral signs. I had copied it in 'Word' using the correct symbols, but it has been through image and forum text editors that do not know Calculus.

I do not know what it tells us, nor if it has any purpose – but that is not as facile as it seems. For it's possible Green derived it within some abstract study, but later workers have found real-world uses for it as science and engineering developed since his time.

It's certainly a candidate for being followed by that mathematicians' equivalent of the Haynes "Re-assembly is the reverse of the dismantling procedure" , the time-honoured, "… from which we can see that…" .

.

So perhaps while mathematics is not determined by physical applications, the applications are certainly determined by mathematics; but in many areas of practical uses of maths, which came first? Was the maths already known as an end in itself, or non-existent until the application revealed it by its own nature?

[Anonymous]

Posted by Nigel Graham 2 on 29/05/2022 22:38:56:.

Was the maths already known as an end in itself…

I suspect that is largely the case. I don't suppose that the mathematicians developing the theory of finite fields in the 19th century had any inkling that they would be central to coding theory in the later 20th century. Coding theory underpins mobile phones among other comms systems.

Andrew

[PatJParticipant @patj87806]

I am deep diving, and definitely way over my head, but it sounds sort of like a chicken-egg arguement.

Which came first, math(s) or man?

Is math(s) just a manmade construct? and doesn't really exist outside the imagination of man (people is the politically correct term these days I guess).

I am sure I will never know the answers to these questions.

Many differential equations can be understood to some extent by understanding that something like dV/dT is a relationship to a rate of change of some variable.

The stylized "S" indicates the sum of an area underneath some plotted curve.

Similar to an equation like pie=radius-squared, except perhaps we want to tie in the rate of change of a rotating radius of a circle, along with the sum of the area of a circle.

In the end, many of the complex formulas break down into some fairly simple functions.

As I understand it, the summation and rate of change variables are transformed into things that can be solved using more standard math functions.

I don't have a great understanding of differential equations, but I do know that many of them are far simpler than they would appear at first glance, once you understand the symbology.

DE can appear rather exotic, but I don't think many of them are really that complex.

.

[PatJParticipant @patj87806]

And one last parting note on this topic.

As I learned in grammar school about how vast the universe apparently is (size is relative I guess), and what a very small part of it we are, I thought of the following statement:

"We are but tiny insects smashed upon the great windshield of life".

Hopefully they won't turn on the wipers until I can get some serious engine building done.

.

[duncan webster 1Participant @duncanwebster1]

And remember, when that fly is squished on your windscreen, the last thing that goes through its mind is its backside

[Martin WParticipant @martinw]

Mathematics has existed since the beginning of time but it is only very recently that we have started to understand the language and we are still learning.

[GeorgineerParticipant @georgineer]

Posted by Andrew Johnston on 29/05/2022 21:06:01:

[Maths] exists in its own right and is independent of any practical application, at least to pure mathematicians. Mathematics is a useful tool, but is not determined by physical applications.

Andrew

That was the attitude of the lecturers visiting from the maths department when I was doing my Electrical Engineering degree. They hated us because we kept sullying their beautiful Maths with real-world applications, and even more so because we used j for the square root of -1, because,as far as we were concerned, i was already taken for current. However, I found their explanation of imaginary numbers very helpful:

First, imagine an xy graph with real-positive numbers on the right-hand of the x-axis.

If you then multiply the real-positive numbers by j you have imaginary-positive numbers. Treat j as a 90° anti-clockwise turning function, and any imaginary-positive number can be plotted on the upper half of the y-axis.

Multiply the imaginary-positive number by j (turning it through a further 90°, so you have multiplied the original number by j² = -1) and you have a real-negative number which lies on the left-hand side of the x-axis.

Complex numbers (with both real and imaginary parts) lie somewhere in the xy plane.

If my half-century-old memory serves, this is called an Argand diagram, after the French mathematician Robert Argand, who discovered (or invented) it. I am sure my old lecturers would be scandalised if they knew how useful it has been to me

George

[GeorgineerParticipant @georgineer]

It's interesting that – unless I have missed it – nobody seems to have mentioned arithmetic, which is concerned with numbers, while mathematics is concerned with anything but. It is the proud boast of many a mathematician that they can't do artihmetic.

If you look at the Greek roots of the word Geometry, it tells you that its origins lie in measuring the Earth.

And it took me until my sixties to discover that trigonometry has to do with the properties of the trigon. What's that? Count down from, say, octagon: heptagon, hexagon, pentagon, tetragon, trigon. It's a figure with three angles…. Oh! It's a triangle!

Now I must find out about algebra.

George

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