## Is There Such a Thing As Math Without Numbers?

One reason I started this blog was because I had asked myself a question: Is there such a thing as Visual Math? I’m talking about a totally visual form of math – where there are no numbers involved (and no verbal language). Is this possible? If there are no numbers (or quantities, or any symbols that stand for numbers) can we still call it math?  Here are a few definitions of math:

Merriam-Webster defines it as “the science of numbers and their operations, interrelations, combinations, generalizations, and abstractions and of space configurations and their structure, measurement, transformations, and generalizations”.

Wikipedia defines mathematics as “the study of quantity, structure, space, and change”.

These references to “space”, “structure”, and “change” certainly ground the notion of math in the context of physical experience.

But math is genuinely abstract, and whether or not it is ultimately derived from earthly experience, it may

or may not refer to the physical world (as in many problems in number theory, for instance).

At any rate, is number really the essence of math?

This is not a new question, as explored in a book by Geoffrey Hellman. I will not try to answer the question of whether number is indeed the essence of math (as opposed to some other concept such as logic, sets, etc.).

I will instead ask whether math can be experienced, expressed, created, performed, manipulated, or acted-upon, using ONLY visual language, and WITHOUT numbers.

Is visual math simply a different way of thinking about numbers than reading and writing equations?

As you can see – I have lots of questions, and no answers!

I look forward to hearing what you think.

### 6 thoughts on “Is There Such a Thing As Math Without Numbers?”

1. To answers your question, of course, there’s lots of math that doesn’t use numbers. Geometry and the like, topology, logic (though that uses lots of abstract symbols). Early math education, like Montessori, teaches math concepts through physical models that are precursors to abstract number symbols. Ancient Greeks considered geometry to be the basic model of math, and so thought of such typically numerical concepts as ratios in terms of diagrams.

Like you I’m interested in there being an alternate form of math that is totally visual, for the purpose of opening math up to people who don’t digest symbols well, and for the beauty and joy of doing it. The power of Mathematica, but without symbols.

I’ve thought a lot about doing the same thing for programming languages. The main thing I’ve concluded, and this applies to numbers and mathematical notation as well, is that it is mistake to think of number math and visual math as living in different universes Number (symbol) math IS a form of visual math, in the literal sense that symbols are something you see. They are just particularly abstract. But because they are in the same sensory space as diagrams, there are many interesting (largely unexplored) points inbetween diagrams and symbols that combine the abstract power of symbols with the visceral holistic power of diagrams. Worth exploring this intermediate land; I believe that’s where the gold lies.

There’s a beautiful series of books called Proofs without Words, worth seeing.

I see you already found the TED talk I would have pointed you to. It’s very interesting to see how they handle things like algebra entirely visually without notation.

• Thanks for the illuminating answer, Scott.

So…if there is such a thing as math without numbers, then we would want to change the definitions from the ones I cited, so that they don’t include references to “number” or “quantity”. And I don’t know if there are any official definitions without these references. Are there?

You made an excellent point in saying that even when we use symbols, we are using a form of visual language, because these symbols are arranged, manipulated, and interpreted visually…which is part of the reason I find this to be a hard question to answer. “Number” is a concept that can be expressed using both math symbols and pictures (and even sounds and movements). No matter how you look at it, it is a mapping from analog to digital.

I still have a nagging question: on the neurological level, when one is thinking “math”, is there not some kind of numerical processing happening? Take the example of the square root of 2, which can be expressed either as a math symbol, an irrational decimal number, or as a picture of a diagonal cutting through a square. The diagonal picture expresses this concept in a very physical and intuitive way.

One might say, “The diagonal (hypotenuse) is LONGER than the sides of the square.” Is “longer” a numerical concept? I guess I would say it is not. But “longer” and “shorter” might still be considered as mathematical concepts – worthy of manipulation, analysis, and generalization to other concepts. If thinking about the sides of a triangle simply in terms of longer and shorter is math, then math thinking does not require numbers ( “countable” symbols).

The question, “how much longer” seems more like a numerical question, because now you need to use a quantity. Then again, you could also draw two parallel lines on a piece of paper with lengths corresponding to 1.0, and 1.414213, and call this the answer. Again, visual, not numerical.

So, maybe you’re right!

2. MacSilver on said:

Jeffrey, you will be surprised to know that there is such a thing as math without numbers. Now of course I am not referring to things like geometry, which also can be greatly expanded as a mathematics without numbers. Nor am I talking about things like the abacus. Now algebra could be considered as mathematics without numbers but that is not what I am talking about either. Math without numbers requires the ability to see and think mathematically rather than compute mathematically which is what is currently being taught.

Part of the issue concerning the learning of mathematics is that numbers for mathematicians are nouns, however, for the rest of the world they are adjectives. In other words, math is just a game and you have to learn the rules to play the game. To figure out if this game makes sense in the real world just ask yourself: What is a number?

All are born with the ability to think mathematically, it is the resurrection of that ability that provides the ability to think and reason mathematically without numbers nor symbols for numbers. This is all part of a course on Cognitive Instruction in Modeling Mathematical (CIMM), were are also developing a Cognitive Instruction in Modeling Physics (CIMP). This type of mathematics focuses on how to develop multiple perspectives for building interpretive frameworks. In many ways, mathematics is the science of the mind and not the ability to compute.

3. Geoffrey Cadman on said:

Interesting debate, I approach the problem possibly In a different way, by way of negative numbers. Euclidean geometry would tell us that negative does not exist only positive numbers exist. Examples are numerous in physics. Freezing point and absolute zero freezing point. Merely a convenient way humans address for convenience where we put the nought. The scale of nought being the point at which water freezes and becomes a solid.

Between nought and one on the line we can make divisions two halves four quarters etc. Try and dig a hole in the ground and you will find a new level which is not negative?