# How can mathematical equations be expressed without equations? [closed]

i.e., without directly stating the equation. My question is about explaining math concepts with art and demonstration.

More concretely, how can I explain $y = mx + b$ or $f(x) = 2^x$ without using those symbols OR a graph? Maybe an animation of insects reproducing exponentially would explain the latter, but can you think of a general way to explain such symbolic concepts?

## closed as off-topic by Hohmannfan, Aify, clem steredenn, Separatrix, Pavel JanicekJun 13 '16 at 9:28

This question appears to be off-topic. The users who voted to close gave this specific reason:

• "This question does not appear to be about worldbuilding, within the scope defined in the help center." – Hohmannfan, Separatrix
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• I guess this very much depends on whether you actually want the art or demonstration to teach people the same thing as explaining the math would, or whehter you just want something that illustrates it in a way that would be recognisiable to people that already know it. – Martine Votvik Jun 11 '16 at 9:32
• I guess that this question is off-topic. Maybe it should be migrated to the mathematics SE? – Victor Stafusa Jun 11 '16 at 11:32
• @VictorStafusa: I think Mathematics Educators SE might be a better match. – celtschk Jun 11 '16 at 16:13
• Here's an example of how to express the inverse square law for intensity of light at a given distance from a point source: The apparent intensity of light from a light source located at position A from the perspective of an observer at position B is inversely proportional to the square of the distance between A and B. I think graphs are probably a better way of getting at human intuitions, but I assume the problem you're dealing with is trying to write this down in a way that doesn't make a novel resemble a textbook (or worse, RPG rulebook.) Therefore, I suggest migration to Writers SE. – SudoSedWinifred Jun 11 '16 at 17:49
• writers.stackexchange.com – SudoSedWinifred Jun 11 '16 at 17:49

It would be trivial to do so, though it might not change very much. Proof Theory is the subdiscipline of mathematics devoted to exploring how one can prove statements using entirely syntactic manipulations such as "Add two to both sides of the equation." In Proof Theory, an expression is typically presented in a Formal Language as a finite string of "symbols" taken from a finite set of available symbols known as the "alphabet" for that language.

You can trivially change the depiction of the symbols without changing their meaning. If you want a bunch of insects multiplying exponentially as a symbol, you can do that. The set of inference rules simply has to change to match the new symbols. We can use Category Theory to demonstrate that there is a morphism that preserves the meaning buried in the syntax.

As for approaches which are less trivial, it will be difficult to do so if you phrase the question "How can mathematical equations be expressed without equations." All of mathematics is a very large domain, and we could literally spend the next 500 years building up such a system and proving that it covers all of mathematics. Meanwhile, mathematics would be evolving and growing, so we would probably never catch up! You might need to reduce the complexity of the task by picking a subset of mathematics that you need. For example, much of early mathematics was proven not with symbols, but with geometry. The Greeks developed a set of proofs known as Compass and Straightedge Proofs which made proofs via purely geometric means. They even identified classes of proofs that could not be made with a compass and straight edge, but which appeared to be true. Many of these are now captured in the study of Oragami, which permits manipulation of the paper in ways that were illegal in compass and straight edge proofs.

In all, you can build the concept of mathematical equations up many ways. Ours happens to derive heavily from the concept of "implication" in First Order Logic, which is a fascinating operator. It is the only operator in First Order Logic that is not defined by First Order Logic but rather left up to the reader of the proof to decide what it's exact definition is. This operator could be thought of as the prototype for all syntactic manipulation approaches in all of proof theory which tie us to symbology.

Beyond that, you really will have to use your imagination, on a case by case basis. For instance, there are some who believe that all will be revealed through some deity. For those people, symbolic proofs may be meaningless, but some vision that we cannot understand may be proof of a mathematical statement for them! Philosophy leaves the door for such thinking wide open.

And, if I may note, I proudly did not use a single mathematical symbol in that entire answer, though I did use a few words which refer to mathematical symbols.

I suppose if I wanted to communicate this is the kind of thing I'm talking about, then giving examples where that function applies might be effective.

If you can only communicate via simple line drawings, and extend that to animations, then you can discuss an orbital rendezvous by illustrating the bodies' motions.

This is useful when the receiving part already knows the math under discussion. You prompt him to recognize it. If it's not known, then giving many unrelated examples would mean "figure out what these have in common". For simple equations like you showed, that would be no big deal. For more abstract notions it might still be possible. E.g. if the meaning is that every X corresponds to a Y, then alternating examples of Xs and Ys might make the recipient ponder in the right direction.

I think, though, that not all math can be illustrated with simple examples of applicability. It gets abstract quickly. This would be the first step in teaching the math notation that will be used.

It is actually pretty simple... see this discussion on "proofs without words" (generally, what it is called when mathematically ideas are expressed graphically. Also, there is this wikipedia article discussing the idea in more detail.

In general, these approaches work best for geometric ideas, however mathematicians are clever and many times can generalize a problem to a geometric interpretation. I have a mathematics background, and especially in my line of work (statistics) visualization is everything.

Among other things, mathematics is an abstract and formal language to describe the real word.

Your equation $y = mx + b$ describes a straight line in two-dimensional space. So you could point to a drawing of a line in two-dimensional space, or a similar physical representation. Of course the abstraction will be lost. A drawing of $y = 2x +1$ for $-5 < x < 5$ in red ink might represent "$2$", "$10$", "red", or "thin line". Perhaps it would help to show different straight lines, some in red, some in green, some on paper, some etched into wood, etc.

So if you try to communicate mathematics without equations, either your audience has to come up with the equations on their own, or it won't be mathematics that gets across.