What is “Mathematics”

[A SmithParticipant @asmith78105]

Wonderful & very entertaining. Thanks chaps.

[GeorgineerParticipant @georgineer]

Posted by not done it yet on 29/05/2022 07:33:45:

The way I finally started working problems was to work out the units first, and if the units all cancelled out to give me the unit I was looking for, then I plugged in the numbers. I still use this method.

A useful tool for this (which I discovered disappointingly late in life) is the "Unity Bracket", which relies on the fact that any number multiplied by 'one' remains unchanged in value. So any conversion factor or complex unit can be rearranged to equal one and then cancelled down as appropriate, and if applied rigorously the answer can be guaranteed to be in the correct units.

If we want to convert miles-per-hour to feet-per-second we can approach it thus:

I don't know off the top of my head how many feet are in a mile, so I'll do that part of the conversion in two parts:

1 mile = 1760 yards, so by manipulating the equation we get

[1mi/1760 yd] = 1, and also [1760yd/1mi] = 1.

Also 1 yard = 3 feet, so [1yd/3ft] = 1 and [3ft/1yd] = 1

And 1 hr = 3600 sec, so [1h/3600s] = 1, and [3600s/1h] = 1

Those are our unity brackets.

To convert 40 mph to ft/sec we do it with unity brackets all through, selecting the 'way up' that will enable us to cancel out the units we don't want:

40 [mi/h] = 40 [mi/h] x [1760yd/1mi] x [3ft/1yd] x [1h/3600s]

Cancelling through:

40 [mi/h] = 40 [mi/h] x [1760yd/1mi] x [3ft/1yd] x [1h/3600s]

which simplifies to (40 x 1760 x 3[ft]) /3600[sec] = 58.6 recurring [ft/sec]

As long as you know the necessary conversion factors you can then go on to convert it into metres/sec, furlongs/fortnight, parsecs/femtosecond and so on, in complete confidence that your answer will be correct (as long as you have doe your arithmetic correctly).

George

[not done it yetParticipant @notdoneityet]

58.6 recurring [ft/sec]

Don’t forget to provide your answer to the appropriate number of significant figures.🙂

[DC31kParticipant @dc31k]

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

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?

It is a good question, but one aspect it glosses over is the issue of time: there are innumerable branches of pure mathematics which at the time they were first written down had no practical application, but in the future have become very important. So, if you study the history of pure mathematics, you would not want to bet on anything never having a practical application.

As just a small example, consider quaternions (4-dimensional numbers). They were written down by the Irish mathematician Hamilton and absolutely derided by some of his contemporaries (e.g. [Lord] Kelvin). Yet in computer graphics today they are very useful.

—

On a slightly different note, the relationship of mathematics to societal norms is interesting to study – in earlier days, the concept of infinity was almost verboten as discussion of it by mortals was considered trespassing on something belonging only to the divine.

[Anonymous]

Posted by PatJ on 29/05/2022 23:55:51:

Which came first, math(s) or man?

The question of whether mathematics is a construct of man, or is inherent the universe, is one which has been debated by mathematicians and philosophers for many centuries, without resolution.

Personally I think mathematics is invented rather than waiting to be discovered. But I also think it is an unanswerable question. If mathematics is an inherent part of the universe we would only know that if it can be observed by a single person with knowledge of mathematics. But it might be that the mathematics exists by the nature of their presence rather than being inherent.

Andrew

[Anonymous]

Posted by Georgineer on 30/05/2022 12:04:33:

…we used j for the square root of -1…

Quite right too.

Although I know the Argand diagram I prefer to think of complex numbers (they're not really that complex) as 2-dimensional numbers. I normally visualise them on the s-plane (Laplace transform) as much of my use of them has been related to filter design and manipulation of poles and zeros.

Seeing them as 2-dimensional fits in nicely with the 4-dimensional numbers described by DC31 (quaternions) and 8-dimensional numbers aka Cayley numbers.

Many years ago I had the idea of using quaternions and Cayley numbers to calculate Fourier transforms, as they should simplify, in theory, the computational units needed. But it turns out that quaterions are non-commutative in multiplication and Cayley numbers are non-commutative and non-associative. Another idea bit the dust.

Andrew

[PatJParticipant @patj87806]

I promised myself I would do something other than blog about math(s) today, but I recall one more story that I find interesting.

In the EE curriculum, there was a ton of calculus; basically every EE subject was saturated with it.

Someone decided that the engineering students should take other courses, such as Physics, that had a strong emphasis on calculus for engineers, and so they changed some of the Physics and other coursed, to be "heavy on math for engineering students".

I took a Physics test, and was very shocked to get an "F" grade on it.

I knew I had the correct answers, so I went to the teacher after class, and quizzed him about it.

He said "Well, it was a circuits problem, and we taught you how to solve those problems using parallel equations". I said "Yes, but I learned current division, and it is much simplier, and so that is what I used".

I also said "There is nothing in your problems that state that a particular method must be used to solve a problem; it only asks for a correct solution, and that is what I gave".

I ended up getting an "A" on the test.

One valuable lesson I learned that day was that you are taught to only be as smart as your professor.

Outsmarting your professor was not really expected or condoned.

The same lesson applies to the corporate working world. No boss wants you to be smarter then them, and I had several that worked diligently to keep me from appearing too productive, or to claim credit for the work I was doing.

I solved the corporate problem by starting my own one-person corporate.

No more boss problems; and I no more glass ceilings that I have to work under.

.

Edited By PatJ on 30/05/2022 21:37:59

[Nigel Graham 2Participant @nigelgraham2]

A former superior of mine told me a slightly similar tale of his university course – and one oddly relevant to this forum.

The test question was to imagine a conical mountain of H height, A angle, etc. A railway spirals down from the summit to the plain, rather like a helter-skelter. A truck weighing M runs freely down it. …. Calculate its speed at the foot.

My manager said there were 20 marks allocated to it, but he made a simple arithmetical error right at the end…. and was given 0 out of 20.

Protesting, he was told gravely, something like "Mr. E—, in the real world the answer has to be correct!"

.

On Argand diagrams and imaginary numbers, though they are above anything I ever learnt (well, was taught…) might these be exemplified by a graph produced by a standard test of electro-acoustic transducers?

These resonate at a given frequency and the test applies a broad frequency run of pure sinusoidal electrical signals with the resonance expected somewhere in the middle.

The resulting graph is of impedance v. conductance, or vice-versa, I forget exactly. For a well-behaved transducer element the trace is a loop; a circle with a near-vertical approach and exit asymptotic to the y-axis. All values are positive but the resonant frequency typically lies half-way round the loop, on [x=something, y=0). It's a bit like the steam-engine indicator-diagram in that it conveys a lot of diagnostic information on the subject.

I was paid just to test the things. The scientists did the analysing, but I did ask one what the curve is. "The real part and the imaginary part" , I was told, as if that answered everything.

Art one point I chanced across a paper on investigating the so-called "wolf note" that can be produced on a cello. Not intentionally, I don't think! The paper briefly described the mechanical bowing arrangement set up on a cello in an anechoic chamber, then set off into the realms of Very Hard Sums Indeed. Among them though, I recall seeing a very similar loop graph, though of mechanical equivalents to the electrical characteristics.

So I take it this is widespread in such fields as vibration and harmonic analysis.

When though was the Argand diagram – if these impedance and cello-note loops are indeed such a graph – invented? Did it pre-date such physics but happened to find itself in such areas?

Edited By Nigel Graham 2 on 30/05/2022 22:49:00

[CalumParticipant @calumgalleitch87969]

Yes, prior to Argand imaginary numbers were just considered to be irreducible sums: 1+2i was just two components and you couldn't really say anything more about them. Argand's contribution was recognising that multiplication by i had a very natural geometric interpretation, and it's this that makes the use of complex numbers as a sort of transform for working with oscillations so powerful.

[not done it yetParticipant @notdoneityet]

I’m quite sure the Fibonacci sequence was around long before humans inhabited the planet, although, of course, it didn’t have a name back then.🙂

[Martin KyteParticipant @martinkyte99762]

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

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

I suspected that was what you had in mind. Appologies for not trawling back through all the posts.

I tend to think of the system of Mathematics as a self consistant set of rules. Regarding your last statement "Mathematics is a useful tool, but is not determined by physical applications", I would go as far as saying it's the other way around and it's the universe that is is constrained by the limits of what is mathematically possibe.

Maths itself is precise. As engineers we get so used to using maths to do stuf that we sometimes mistake the application of Maths for Maths in it's pure sense.

Pi is exactly the ratio of the circumference of a circle to it's diameter. That is the maths. Our application of Pi is always an approximation albeit to whtever precision you like. Pretty much any mathematical model of the real word is the same.

regards Martin

[Frances IoMParticipant @francesiom58905]

I prefer, even as an engineer though I did train as a physicist, that Mathematics is a pure human invention based on axioms – however another invention of the human mind are models of the universe – the more useful of these models are based on a mathematics which can be used to produce expected properties of the model universe – in as far as those properties correspond with measurements we accept our model as a valid description – in some ways we are behaving in the same manner as the ancient Greek method of exhaustion to estimate areas of arbitrary objects by determining polygons that bound the object.

Edited By Frances IoM on 31/05/2022 09:56:50

[Nigel Graham 2Participant @nigelgraham2]

Isn't that last technique – for areas – something of a forerunner of the Mid-Ordinate method of Integration (which I recall vaguely as summing rectangles of [Area = Width X Centre-line Height] ) ?

[Martin KyteParticipant @martinkyte99762]

Posted by Frances IoM on 31/05/2022 09:56:04:
I prefer, even as an engineer though I did train as a physicist, that Mathematics is a pure human invention based on axioms

Edited By Frances IoM on 31/05/2022 09:56:50

Hi Fraances

Certainly the notation and language of Maths is invented but the Maths is maybe discovered? Hard to argue that Primes were invented.

regards Martin

[SillyOldDufferModerator @sillyoldduffer]

Posted by Martin Kyte on 31/05/2022 12:04:39:

Posted by Frances IoM on 31/05/2022 09:56:04:
I prefer, even as an engineer though I did train as a physicist, that Mathematics is a pure human invention based on axioms

Edited By Frances IoM on 31/05/2022 09:56:50

Hi Fraances

Certainly the notation and language of Maths is invented but the Maths is maybe discovered? Hard to argue that Primes were invented.

regards Martin

I see Mathematics as a tool used to promote clear reasoning in support of practical and theoretical problem solving. It's a broad discipline – whilst engineers mostly only use maths to extract information from data, scientists often use it to explore and validate new concepts.

As maths is a tool it must be man made.

Maths and the properties of nature it can be used to describe aren't the same. Maths serves the people who invented it and although number and formulae can predict and quantify nature, nature doesn't require maths to do its stuff. For example, although Maths can be used to quantify Forces and predict how they influence events, maths can't define what a Force actually is. Despite humans knowing masses are weakly attracted to each other, and understanding the relationship well enough to land a robot on Mars, we don't know what gravity is. Nature just happens to obey laws that we can describe mathematically, and therefore model nature.

As Maths is a tool it follows that people can and do live happily without it. However. I'd argue it is the most powerful tool ever invented and therefore worth learning as far as our limited time and aptitude will allow. Basic engineering may be valuable, but Maths is absolutely essential to move science and engineering forward.

Dave

[Frances IoMParticipant @francesiom58905]

as a one-time professorial colleague used to point out “there is nothing so practical as a good theory”.

Edited By Frances IoM on 31/05/2022 21:04:08

[duncan webster 1Participant @duncanwebster1]

Posted by SillyOldDuffer on 31/05/2022 20:15:03:

 

……….whilst engineers mostly only use maths to extract information from data, ……..

Dave

Where did you get that idea from? I used maths to model all sorts of things, including solving differential equations. Needs lots of icepacks for sure, but invaluable. I'm not pompous enough to quote the 'no maths = no engineer' so beloved of academics, but some engineers certainly do clever maths, much cleverer than I would dream of

Edited By duncan webster on 01/06/2022 00:11:33

[SillyOldDufferModerator @sillyoldduffer]

Posted by duncan webster on 01/06/2022 00:09:16:

Posted by SillyOldDuffer on 31/05/2022 20:15:03:

 

……….whilst engineers mostly only use maths to extract information from data, ……..

Dave

Where did you get that idea from? I used maths to model all sorts of things, including solving differential equations. Needs lots of icepacks for sure, but invaluable…

The extract information idea can certainly be challenged: it's a generalisation, and they're always built on sand.

My experience of maths is of levels:

• We all manage money to some degree:
  • Doing ordinary shopping, I have a rough idea what I can afford, which is why I'm more likely to buy Beer than Champagne.
  • Buying a house, with a wife and young family on the way, I did a much more careful analysis, considering my reliable salary, low likelihood of generous pay settlements, and promotion prospects relative to inflation and interest rates over a decade. It led to a marital dispute, because the figures revealed my desire to prioritise the house, collided with my wife's urgent desire to fill it with carpets, cushions, and Tippets for the children.
  • At work, even more elaborate financial analysis was done before investing corporate dosh in equipment and projects. For example, Business Cases always required me to provide a Discounted Cash Flow. A DCF compares Benefit – Cost over the lifetime of a proposed investment with the return that would be obtained by putting the money in a safe Building Society and just taking the interest. (Today's ultra-low interest rates justify UK government's distinctly un-Conservative spending policy, and a cynic buying a house today, might worry the politicians might be planning to push the value of the debt down by allowing a risky burst of high-inflation, hoping they can control it.)
• Engineering is closely related to money in that 'Any fool can do for a Guinea, what an Engineer does for a Pound'. Much more to engineering than bolting stuff together to get a result, especially when more than one option is available. But it too has levels:
  • Given walls of 'n' square metres, how many litres of paint to I need to buy? Or. how much electric cable is needed to run a spur from the Consumer Unit to a garden workshop, after noticing Manhattan Routing the wire around the outside of the house is least work and intrusion? Fairly simple – extracting information from data .
  • If 50,000 litres of water per hour are to be pumped from a 50 metre deep well, how powerful does the pump need to be, and how much pressure will the pipes have to withstand? Moderately complicated, especially if the cost of high-pressure pipework is reduced by pumping in two or more stages, but still extracting information from data and using it to select off-the-shelf solutions..
  • If a nuclear reactor produces 100MW of waste heat, what combination of pipework, coolant, and radiator provides the cheapest fail-safe way of managing it, ideally in a profitable way? This example is much more complicated: the coolant could be air, water, Carbon Dioxide or molten Sodium, each with it's own peculiar set of advantages and disadvantages. There are multiple answers, and I think the only practical way to compare them is with mathematical models, Aerospace and many other design activities call for a similar approach.

Engineering maths can be and often is highly advanced. For example, it was once fairly easy for an amateur to hot up a car engine by haphazardly opening up anything that impeded the flow of fuel into the engine, and exhaust out of it. Bigger carburettor(s), supercharger, chamfered ports, wide bore exhaust, silencer removed, pipes polished internally etc. Such no-sums-necessary techniques get significant improvements, but doing better requires deep thought and analysis. For example, early IC's engine designers worked hard to implement the maximum efficiency Carnot Cycle. Clever stuff, but modern engines are polytropic, balanced to optimise several other desirable features such as acceleration and reduced pollution. In short, getting the best out of an IC engine requires a solid understanding of mathematical thermo-dynamics beyond me and most petrol-heads. The modern motor car has been optimised in many other ways too, from road-holding suspensions to absorbing crash energy by folding up gracefully. The day of the individual inventor developing a successful mechanical product in a shed has pretty much gone.

Practical engineering is vital it's own right – it's how things get done, demanding skill and experience academic types rarely have. I admire Alec Issigonis and the chap who serviced and MOT'd my daughters car yesterday. They both do technical stuff I can't!

Scientists go a step further: the maths that led to E=mc squared, Black Holes  and Dark Matter are exploratory, rather than Information from data.

Dave

Edited By SillyOldDuffer on 01/06/2022 11:06:34

[VersabossParticipant @versaboss]

An interesting discussion about something I don't understand very well. Anything higher than differential calculus was too much for me back in my schooldays, long ago…
However, for unknown reasons rooting back in time I found in my stack of unread/partially read books the following one:
The Mathematical Experience, by Philip J Davis and Reuben Hersh. Wikipedia says a lot about these two mathematicians. The book has an interesting comment by the New Scientist on the front page: "An instant classic, it deserves to be read by everyone with an interest in the future of the human race". Oh well, maybe I should read more than just a few pages, but it is quite hard work for me. And my "interest in the future of the human race" is maybe not so high now.

But what bears reference to this thread is the beginning of the first chapter, which has the title "What is Mathematics?" This is followed by what the authors call a naive definition: "Mathematics is the science of quantity and space".

Thinking hard about that,
Hans

[PatJParticipant @patj87806]

Posted by Versaboss on 02/06/2022 22:13:04:

But what bears reference to this thread is the beginning of the first chapter, which has the title "What is Mathematics?" This is followed by what the authors call a naive definition: "Mathematics is the science of quantity and space".

I would agree with that definition, but push it a bit more to say it is also about motion, and related things such as velocity, acceleration, etc.

One part of mathematics that is useful is the "bean counting" aspects of it.

Another part of mathematics is how it can be used as a predictive tool, assuming your model is sufficiently accurate. With the correct mathematical model, on can predict the behavior of things that one has never seen, which is quite magical in some respects, and very profitiable if you run an engineering firm.

.

[HopperParticipant @hopper]

Posted by Frances IoM on 31/05/2022 21:02:44:
as a one-time professorial colleague used to point out "there is nothing so practical as a good theory".

Edited By Frances IoM on 31/05/2022 21:04:08

And another great man said : "In theory, theory and practice are the same. In practice, they ain't". (In theory said by Albert Einstein. In practice, attributed to various, ranging from Richard Feynman to baseball coach Yogi Berra.)

If you really want to know what maths is, a good place to look is the philosophy of mathematics. Wiki has a succinct (not!) summary here **LINK**

The greatest human minds have wrestled with this question since the days of the ancient Greeks and come up with multiple theories of what maths is (are?? Hmm??) In practice, it seems nobody can really say completely or for sure what it all is.

Now, one scientist once said that scientists need the philosophy of science about like birds need ornithology. But one would hope that scientists, including mathematicians, have a greater level of self-awareness than the average bird. Although, sometimes I am not sure…

[duncan webster 1Participant @duncanwebster1]

The 'scientific method' is to derive a theory from first principles, but to then test it by experiment to make sure it properly describes the real world. If it doesn't then either the theory is wrong or the experiment is wrong. Usually the former, but sometimes the latter. Working out for example how much a beam will bend is based on theory, but it is well supported by long years of use.

Is looking up a formula and plugging in the numbers mathematics? Arguably it's arithmetic.

Why are some people, including those in positions of power, proud of being inumerate? Let's set the cat amongst the pigeons, accountants don't do maths, they do arithmetic. Same applies to the majority of the population. Why then do we complicate the issue by teaching schoolkids the clever stuff rather than life skills like not taking out loans at usurious interest rates. Some of the stuff my kids were taught was simply wrong, for example one question in an exam paper had one equation with 2 unknowns and they were asked to find both unknowns. The maths teacher would not have it that there were an infinite number of answers

[SillyOldDufferModerator @sillyoldduffer]

Posted by duncan webster on 03/06/2022 12:46:28:

…

Is looking up a formula and plugging in the numbers mathematics? Arguably it's arithmetic.

Why are some people, including those in positions of power, proud of being inumerate?…

Duncan is pulling my leg with 'inumerate', he knows I'm innumerate!

He's right though, the number of people I meet who are positively proud of not knowing things is scary. Worse, lots of people think ignorance of technology means they are superior beings, describing those in the know as 'mere technicians'. Not just their ignorance that's alarming, it's the faulty logic and self-delusion that goes with it! How many Prime Ministers would you trust to return home with the right stuff after being sent shopping?

I argue 'Arithmetic; is a member of the Set 'Mathematics' and is therefore respectable!

Bertrand Russell showed maths had limits with paradoxes such as 'Is the set of all sets a member of itself or not?' In Doctor Who, asking this question reversed the polarity of the neutron flow and the resulting infinite recursion caused the Big Bang. This is why Health and Safety is paramount in my workshop – who knows what this Clown will do next…

Dave

Edited By SillyOldDuffer on 03/06/2022 13:50:58

[duncan webster 1Participant @duncanwebster1]

Being innumerate is even worse than being inumerate, but it shows you can spell, which I plainly can't!

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