Mathematical/logic proof that unicorns exist.

@ArishMell As soon as you start doing anything with math you need sets and set theory. Statistics, analysis, number theory, ... It's closely related to logic itself too.

@Luke73 I'm used to seeing applied mathematics associated with science and engineering, and none of that used sets. It is mainly equations and formulae, a lot of them trigonometrical; series, calculus, logarithms; but I can see set theory would have applications in statistics.

@ArishMell For example to define a function, you need to have sets. You need to have sets for all the basics.

@Luke73 I was being a bit tongue-in-cheek; but as I say, none of the professional mathematics I saw ever used sets.

I suppose because it was all tools for specific, real purposes like designing, testing, measuring, analysing physical things, not statistics. Let alone Pure Maths for maths' sake. Mostly far above my level but nothing unconventional; not a Set in sight.

I saw a report containing equations prefixed with several definite-integrals signs like a stylised swannery; and a text-book on stresses and strains in pressure-vessels. Also a Pure Algebra paper offering an alternative solution of someone-else's equations, full of symbols unknown to me - and probably of no practical use! My Dad's old engineering text-books are full of extraordinary maths, especially thermodynamics and steam-turbine design further complicated by Imperial, not metric, units. Still not a Set in sight.

(I needed use only fairly simple formulae, with occasional trigonometry and logarithms. My superiors handled the difficult maths.)


A Maths course I took introduced Functions: fancy equations, not wrapped up in Sets but still Very Confusing with their "many-to-one-and-one-to-many" antics not defining how many of what!

It also had Matrices, too abstract for me. One of my superiors, trying to help me, explained they are used in engineering work such as hers, Finite Element Analysis, involving gigantic blocks of simultaneous equations. Well, she did have a PhD in Mathematics! I baffled her by thanking her for her help, but adding I was still lost. Mathematicians cannot understand why other people cannot understand mathematics.

I have not studied Statistics, though; so admit I may well have not met real uses for set theory.

I know there are also Professors of Mathematics who, when not supervising PhD students or contract sum-smithing, love research in "Pure Mathematics" - abstract number-puzzles too arcane to be sullied by mere trade! (Perhaps Prof. Charles Dodgson thought no-one could so despoil the Matrices he developed - when he was not writing his other fantasy, that for children about a girl following a rabbit.)

@ArishMell Usually set theory or sets are not called like that, but for example you have image, preimage, space, subspaces, ... those are all examples for certain sets. That's common and that was for a long time a problem because a set was never really defined. You often implicitly use sets. For addition and subtraction you need to define a set for. And that's what algebra is about, group theory which works with sets. I don't think there is even a discipline in math that doesn't use sets.

@Luke73 What is Set Theory actually for?

As I say, I have never seen the word "set" attached to any mathematics I have seen, from the simplest three-value equation like A=l.b to deeply arcane technical calculations used for particular fields of work.

All these use algebra, obviously, as there are very few mathematical relationships that can be expressed without it, or use only a few simple formulae. Those exceptions include -

- Euclidean Geometry (totally non-numerical),
- graphs of values not numerically related, like temperatures or sales figures not needing calculating,
- basic finance (more arithmetic than mathematics), and
- converting values between Imperial and ISO units (simple multiplications by constants).

Otherwise, algebra is maths' language.


My background:

I accept Sets have a place, but it is a place I have not personally encountered beyond an introductory level in an experimental "School Mathematics Project" (SMP) syllabus at school; alongside what it sniffily called "traditional" (i.e., useful) maths.

Infants' School (2 years): addition, subtraction, the times-tables, simple multiplication and division, simple money-sums.

Primary School (4 years): Long Division and Multiplication, Fractions, Compound Multiplication. (The last was prices of quanties of commodities costing £so-many in bulk; with £.s.d currency and Imperial units of mass.) Difficult enough but still no "Sets".

Secondary level (5 years course): More advanced arithmetic (HCF, LCM, %s, money-interest, logarithms as calculators), algebra to simultaneous and quadratic equations, graphs and basic calculus, mensuration, trigonometry, Euclidean and numerical geometry. All in one syllabus - still without Sets.

Work (46 years): various workshop and lab-floor level, science and engineering related, roles. In this, and to some extent my hobbies, I saw lots of very advanced mathematics indeed - not to use myself but at least I am aware of them. I do seeing no use of the "S-" word, and little of the "M-s" word.


As an adult student in evening-classes, I took the present, standard and "Advanced-Level" school-syllabus courses for work reasons. It was considerably reduced from my school-days equivalent, but all real-word maths. No airy-fairy SMP. For a career in digital electronics you'd presumably learn its own maths with that trade.

Neither course mentioned Sets. Nor did their text-books. They did introduce the mystic Matrices- on their own, without meaning, purpose or link to other mathematics.

No Sets in my father's Degree-level maths and engineering text-books. Nor Matrices, but one describes something called "Determinants" for solving simultaneous equations of >2 unknowns. As a Chartered Electrical Engineer, much of Dad's work would have been intensely mathematical.


SMP tested a potential syllabus for the dawning Age of The Computer - mid-1960s. No algebra, as I recall! Transform Geometry, "nets" (academics' name for "developments") of regular polyhedra, Boolean Logic, binary and octal arithmetic. And, Sets! On their own, without meaning, purpose or link to other mathematics.



Your definitions tell me Sets are only Pure Maths abstractions trying to justify themselves; fun academically but pointless for most real-world purposes.

I dare say Sets have certain special uses (in computer-programming and processor design, and some ares of Statistics, perhaps?). However, if you learn and use maths for practical purposes otherwise, you don't have the need, inclination or time to further complicate work-a-day algebra with academic abstractions!

@ArishMell Set theory describes what sets are and the basic operations you can do with them. It started because for a long time there wasn't a definition for sets and it was used naively, which lead to some paradox, like Russell's paradox that I referenced in my post.

Again what you mentioned is something mathematicians struggled for a long time with, they used sets intuitively. That works most of the time good enough.

You brought up geometry, that's all about sets. What is a line? A set of points. What is a square? A set of points.

Graphs are usually a function. And what is a function? A relation between two sets. For example you observe the temperature in a specific time frame. Your domain is the time in that time frame. A domain is a set. And the temperature is the image. Another set. And the graph or function or how you want to call it, is just a set of pairs. The first one is the time and the second one the temperature. You probably didn't view it like that but that's how it basically works.

In any more advanced mathematics courses, the first thing you learn about are sets.

If you want a "real world" application, you can for example look at groups of people, people who have brown hair, tall people, ... and you can look into if they share other common traits, etc.

I'm not a linguist but often you use other words instead of the word set, like interval, range, numbers, ... They are all there whether you want to see them or not. Set is just the name we've given that specific concept.

@Luke73 Thank you. Yes, I understand that, but I have never seen any mention of sets or need for them in any technical mathematics, even at levels far above anything I could have attained, hence my concern that about needless use.

Even when used as in your example of a square joining a set of 4 points, the use was purely colloquial, just linguistic convenience. Similarly with graphs: no formal use of "set" beyond a basic comment that it's a set of co-ordinates. Not diving into Set Theory where it's not necessary.

@ArishMell It’s taught and used in any course at higher mathematics at university. I know people from different fields and people who teach it too. For example in calculus, you can’t do it without it. For example for the intermediate value theorem or for splitting and combining integrals.

And no, typically a square contains more than four points. In the real plane a square contains infinite many points. So for example when you check if two figures intersect, you check if the union of these two sets is non-empty basically.

Another example where it’s necessary to understand sets is when making multiple graphs in one figure. Usually you want the graphs to share an axis, so you want the domains to be the same, or at least a subset of the biggest one.

@Luke73 Thankyou for the explanation but the more you show its subleties, the more Set Theory seems a mathematical analysis of maths for its own sake.

I am used to using simple maths, and came into contact with far more advanced mathematics - including calculus to a high level - and none of it uses sets. I can see you can describe a graph or a geometricall figure or perhaps an equation in terms of sets, and can see it being relevant to statistics almost by definition, but it is not necessary to learn Set Theory to learn most mathematics. Let alone to use mathematics for anything practical, whether as simple as the area of a circle or in very advanced engineering designs.

Instead all the maths I was taught - and all the books I have - go straight to the point. They don't battle through thickets of sets for everything, not even calculus taught from First Principles.

I had no idea Sets are even involved in the diagonal of a square or a complex harmonic analysis! You just dive straight in with the appropriate equations - though I know harmonic analysis does involve fiendish trigonometrical and logarithmic series.

+++++

This reminds me slightly of when I used to contribute to a site called "Answers.com", part of Wikipedia. (It closed then later re-opened but by subscription, which I refuse on principle though I sometimes donate to Wiki. itself.)

Answers is or was a classified Q&A knowledge site on a huge range of topics. The Mathematics included many Americans trying to learn US / ISO measurements conversions. Already used to metric measures, I helped two groups of questioners.

One was householders calculating the doses for their private swimming-pools measured in feet, of disinfectant sold in metric units with metric instructions. I walked them through the arithmetic, step by step. Some did not realise you need the volume of water, not its surface area.

The other appeared to be school-children trying to learn the basics: feet to metres, miles to km, gallons to litres. I would not help them cheat by just giving the answers as some respondents did. Instead I also told them how: look up the appropriate, widely published conversion constant, and multiply accordingly. Just simple A = B X conversion, "times sums".*

The homework questions were spoilt by two characters making its as baffling as possible; probably deliberately. They did not use over-analysis as your beloved Sets do, but strange over-conversions[/]. They used the word "Algebra" but no algebra itself, and referred to a text-book on "Dimensional Analysis", which this is [i]not. They often made mistakes in their own sums, too! E.g., Miles to km? So miles to inches to centimetres (non-"Preferred" units anyway), back up to km.

.........

*(I ensured I used US, not UK, Gallons. For miles to km I suggested 8/5 is usually simple enough for mental arithmetic, and sufficiently accurate for most real journeys.)