Help Needed: Concrete Analogies for Abstract Math Concepts

[Dragonfall]

I'm giving a talk entitiled "concrete analogies of abstract concepts", where I give examples of concepts in mathematics that might have arisen in things found in every day life. I already have "chess" for isomorphism and "acronyms" for homomorphisms, but I'm running a bit dry. Anyone got any ideas?

 

[mathwonk]

rotations of cubes give examples of group operations and also of homomorphisms if you look at the action on axes, or other subsets.

in fact it is my oopinino tha essemtiwlly abstratc concepts arose from concrete examples, so it should be possible to find them for anything.

abstract divisibility of integers arose in Greek times as the theory of measurement. I.e. they said one unit "measures" another instead of divides it, which clearly means that a meter stick can be used to emasure precisely those lnegths whose length is divisible by one meter.

indeed the familiar proof that all ideals of integers are priniple folows fro this concrete analogy a follows:

the usual abstartc proof says to atke an element of the dieal of shortest length nd them use the divison algorithm and contradiction to show it divides all other elements.

concretely one looks at any finite collection of elements on the line and then consider two other lengths, namely the shortest length that can be measured using those given oens as measuring sticks, and second th longest length that can be used alone as a measure to measure all the given ones, assuming these exist. they exist of course if and only if the given lengths are "commensurable" which the greeks initially assumed always occurred.now we claim that given two commensurable lnegths on the line, that the two lengths are the same, i.e. the longest length that will measure both of them, rquals the shortest length they can measure together.

to see this consider all points of the line they can measure together (i.e. all integer linear combinations of the two given integer multipels of some "unit" length).

so you are given two measuring sticks and using them and some given starting point, draw in all points whose location can be exactly measured from that starting point using those two sticks.

assume there is a shortest one. now we claim that all those points are equidistant, i.e. they are all multiples of that on shortest one.

to see this note that if two points can be measured so can their difference and sum. hence if the distnace between A and B were shiorter than tht betwen P and Q, we could measure off the shorter distnace betwen A and B and then ad it to P to get somewhere between P and Q.

Now that all succesive pairs of points are equidistant, they must all be multiples of the segment from the starting point to the first new point. hence that shortest distance that both sticks can measure, is also the longest length that will measure all the others, including the two originally given ones.

so the abstract proof that every ideal of integers is principal is equivalkent to the geomnetric fact that the shortest distnace measurable by two measuring sticks, equals the longest distnace that will measure both of them.

the conceopt of a "neighborhood" I am topology is the same as the commonly used one in everyday speech. once a builder friend of mi e said he had made someone an offer, "between 475 and 500, somewhere in that nighborhood" then he remarked to me it was not a very large neighborhood, so i told him that as exactly what the term means in math, that one srudies things precisely, by repeatedly approximating them, and in fact one specifies smaller and smaller neighborhoods of the desired value.

 

[mathwonk]

linearity is familiar proeprty, as is exponentiality. in making cookies by a recipe, we assume that doubling th recipe, i.e. doubling every ingredient will doublt the output.

but in buying wine e.g., if robert parker gives it a score of 100 it sells for 10 times or more what it does if he scores it 90. that's not linear pricing that's exponential.

so unless your enjoyment of a 100 point wine is really 10 or 20 or even 100 times as great as for a 90 point wine, you are foolish to buy one purely for drinking pleasure. reduction mod n, a trick for shpowing certain equations have no integer solutions, is reflected in the arithmetic of "clock time" since going around the clock n+m hours yields the same time as going around for n hours and then for m hours more.

 

[mathwonk]

the sensation of hearing loudness is apparently not linear in relation to the number of decibels of the sound. this is another exponential relation.

I believe Fourier series for representing arbitrary functions in terms of periodic ones, are analogous to the composition of pure sound waves to generate more compklex ones, similarly for light, but the physicists need to be consulted here, such as helmhlotz.

addition of velocities, which is approximately linear at low ones i guess, is audible in the doppler effect when a siren wailing police car passes at high speed. i.e. if velocities did not add there would presumably be no effect.

second derivatives of position are felt in a car as acceleration forces.the concept of two sets having the same size or being bijectively equivaklent, is distinct from being able to count them. this is exhibited in the story of th cyclops in ulysses who matched up his sheep against poebbles in a pile, to be sure they all came back in, after being blinded, but he could not count, hence did not know "how many" sheep he ahd.

he jiust knew if the same number went out as came back. if he did not save the pile overnight, he would not miss one that as eaten by ulysses's men inside the cave at night.

 

[Dragonfall]

I'll try to work some of that into the talk. However some of your suggestions are still too "abstract". I want this talk to be as if I'm writing a pop-math book.

 

[Gib Z]

How about a combination of maths and physics? Mechanics is a great example, try something about cars traveling at high velocities or something. Maybe that if a car A is traveling twice as fast as car B, car A actually takes 4 times as long to stop, not twice as long.

Or how about the classic example of a magical ball where you drop it from say, 1 metre and everytime it hits the ground it bounces back to half its previous height. Ask the audience if they think it will ever stop, and what distance such a thing would travel that wouldn't stop. Then bring into relation the series that describes such a behaviour, they may be surprised to hear that an infinite number of terms can sum to a finite amount.

Perhaps a poker game, where a lot of complex probability is needed to accurately describe. Or maxima/minima problems for manufacturing companies deciding how to cut out their cardboard from a rectangular piece to get the largest profit.

 

[quasar987]

Is this the kind of thing you're talking about?

Plz don't laugh. Here's an concrete analogy for expressing the idea of an Archimedean field.

"You can imagine a field as the body of an animal (i like to think of a cameleon-like reptile) and the natural numbers in it make up its spine. A field that is archemidean has no deformity, whereas the spinal colum of one that is not Archemedean curl up on itself at one point (such that there are points in the body of the animal that are greater than any integers)"

 

[Dragonfall]

Well, sort of... but I'm really thinking of something analogous to my former analogies: two chess sets, one marble, one wood, but they play the same game, as isomorphism; acronyms, where abbreviations mean loss of information, ie CSIS would have a preimage containing "center for strategic and international studies" and "canadian security intelligence service" and even "computer science and information systems".

 

[fopc]

Well, sort of... but I'm really thinking of something analogous to my former analogies: two chess sets, one marble, one wood, but they play the same game, as isomorphism; acronyms, where abbreviations mean loss of information, ie CSIS would have a preimage containing "center for strategic and international studies" and "canadian security intelligence service" and even "computer science and information systems".

Rather than use two chess sets that happen to have pieces made of different material
as your example, why not demonstrate that two apparently different games are
actually related by an isomorphism? (The two games are identical in structure.)

Read here for an example.
http://j-paine.org/students/lectures/lect6/node9.html