# Computers in a society whose mathematics is limited to geomety

Consider a society whose mathematics never developed numbers. Instead, the entirety of their mathematics is highly developed geometry (the kind of work done in the first few books of Euclid's Elements).

It is possible to use this type of geometry to simulate algebra, but it is awkward, meaning that many results that require efficient manipulation of quantities would never be discovered. Most of the most basic things, however (such as arithmetic and solving simple equations) would have analogues in their number system.

I don't expect that any pre-1800s technology would be completely out of reach for this civilization. However, in 1837 we see the first design for a general purpose computer: Charles Babbage's Analytical Engine. Charles Babbage was a mathematician, and it takes a jump of mathematical abstraction to imagine how a general-purpose programmable computer could be created.

How would a civilization whose math was based on geometry develop a similar prototype for a computer? If they were capable of achieving this, what would their computers look like?

• Books 5, 7, 8, 9 and 10 Euclid's Elements are about numbers and arithmetic. The Classical civilization developed number theory alongside geometry; Diophantus, for example. Descartes lived long before Babbage. As for numberless technologies, have your ever considered commerce and accounting? Both are quite hard to do without numbers... As is engineering without calculus. You may want to explain what "math[ematics] based on geometry" means. Apr 23, 2018 at 6:51
• The whole premise is grossly faulty, because animals have been counting for tens of millions of years. livescience.com/61084-can-animals-count.html cnn.com/2015/06/10/us/chimpanzee-crow-intelligence-studies/… Likewise, you've got to know how many goats you own, how large your harem is, how many children you have, and the king needs to know how many sheep you have so he can tax you, and the Pharaoh's vizier needs to know how much grain has been harvested and stored so the people don't starve during the 7 year famine. Apr 23, 2018 at 7:08
• Synaesthesia might be a way to do this. There are people whose mind substitutes shapes and colours for numbers. They are still doing math with numbers, mind (and much faster than normal people do). So this is a bit cheating. Apr 23, 2018 at 8:00
• Distinction between one and many is very basic. I don't think you can avoid it, and from that point numbers will simply happen. Apr 23, 2018 at 9:09
• I don't think this is remotely viable. What would help is an example of purely geometrically based mathematics, especially simulating algebra in the context of commerce, for example. Apr 23, 2018 at 13:44

Although it is highly unusual to develop computers without numbers, it may be possible to enviseage a civilisation that uses an early computer to solve other problems.

There are close developments to this in history - for instance many ancient civilisations had a fascination for time and astronomical observations, and developed Stonehenges and other 'structural' tools to calculate positions of stellar objects, planets and predict events. By placing stones geometrically, these could be used to predict events over time, an early tool to solve a problem.

Other tools allow geometric problems to be easily calculated, such as compasses and rulers - before the advent of 'CAD' software we used geometric tools to solve geometric problems, such as compasses to find bisectiing lines, proportions in lieu of angles. This is for instance useful methodology for quickly finding lengths of ramps given certain gradients, perpendicular angles given certain directions. It was not unusual for me to do the above without making reference to 'numbers'.

It is possible to combine these tools into a computer to form your first 'geometric computer' without first discovering, say, an abacus (which is one of the first counting computers). Such a tool, to fullfill the definition, must be some way of developing a result, such as an angle, to solve a problem given certain variables, for instance a height drop in an aquaduct over its length. Practical problems found in real life that need solving such as aqueducts, roof pitches, wall heights, foundation depths, would be the origins of these problems requiring every-day solving.

I would imagine many of these simple types of computers to be the origin of your non-number computer. Over time, it may be possible to expand the capabilities of these simple computers to fullfill more complex functions, similar to the Antikythera computer, using geometrical mechanisms to solve when a future event may occur.

• It's hard to make those gears without numbers and arithmetic... Now seriously, what you describe and don't know how to call it is an analog computer. Analog computers were first developed at the beginning of the 20th century to solve problems related to targetting naval guns, and from the developed strongly for more than half a century, with particular strength in applications which required solving differential equations. It wasn't until the 1960s that numerical computers were able to duplicate the functionality of analog computers. Apr 23, 2018 at 15:36
• ... But the very problems where analog computers were applied depended on an understanding of calculus. Cannot really have calculus without arithmetic and algebra. Apr 23, 2018 at 15:37

With geometry you can create something that looks like a computer, but without the numbers it's just an Etch-a-SketchTM

A "computer" based, not on numbers, but on geometry alone is capable of only one thing: geometry. It could draw circles and lines at intersection points that permit the expression of angles, but you can't do anything more because there isn't anything more. Technically, it can't even draw circles with a given diameter because, while you can know that one circle is larger than another via geometry, you cannot know how much larger, and therefore cannot draw a circle with a precise diameter.

How would such a "computer" work? Not unlike an Etch-a-Sketch where, by trial and error, a series of basic gears and levers would drive the drawing knobs for the purpose of drawing circles and straight lines.

This, of course, isn't a computer as nothing is computed. It's nothing more than an underdeveloped plotter.

It would be wrong of me to say computers aren't difference engines. That is the result of what they do. Even our modern CPUs are little more than very capable difference engines. But the reason they work, the basis of their methodology, is fractions.

The difference between the count of teeth on two gears is a fraction. The difference between the diameter of two gears is a fraction. These two concepts were fundamental in the development of Babbage's analytical engine. He certainly stood on the shoulders of Euclid when he created his machine, but his machine could not have existed without numbers, and more importantly, fractions.

If you think about it, you can use geometry to divide a circle into halves, thirds, quarters, etc. But that has no intrinsic meaning if you can't express the idea as 1/2, 1/3, 1/4, etc. How do you create a computer that can divide 77 into 99 if all it can do is draw circles and lines? Eventually it needs to count the number of pieces it has.

It's hard to imagine a society that cares about geometry but doesn't care about running numbers

And by "running numbers" I mean the black-bag work accomplished by mugs in a mob who are taking the poor for their last dime in an illegal lottery.

It could be argued that modern mathematics descends solely from humanity's intense desire to know what time it is (I'm getting to running numbers). Geometry was certainly involved because arcs and lines were used to deliniate the passage of time in its simplest way: the passage of a shadow. But you can't use geometry to express what time it is. That requires numbers.

Of course, once employers knew how to tell time within reason they had the ability to pay their workers in proverbial bushels of wheat per-increment-of-time.

And that meant that some poor schmuck had to keep a record of how many bushels of wheat had been paid to the workers and how many bushels-per-increment were being lost to unproductive workers and gee if I just happen to smudge that number right there I can take an extra bushel for myself.

And that meant that his friend, who happened to know a couple of big honking Tongans, could lean on his I-keep-an-accounting friend (let's call him a "bushel counter" or a "bunter" for short) to get a few extra bushels for himself.

And that meant, obviously, that even more bushels could be had if we set up a raffle to win this handy-dandy flint-and-iron set if you just put a cup of wheat into the pool...

Wait... how many cups of wheat are in a bushel?

Ah... Calculus!

Conclusion

Geometry would let you create a machine, but without numbers (especially a knowledge of fractions) you couldn't do anything useful with the machine except draw more circles and lines.

I'll give a brief overview of what is possible and then try to construct something viable or at least give further research directions as an actual answer.

# Geometry

Well, geometry is a surprisingly broad topic. The classic euclidean geometry is more than 2000 years old, Ancient Greece, and so on. However, there are other geometries that are mostly brainchildren of 19th and 20th centuries.

While there are some classical non-euclidean geometries, I have a feeling that a better explanation is the following. A large part of geometry studies the relations of lines and points and surfaces and so on. So, we could, for example, construct some example in a usual euclidean geometry and than use a mapping that makes points from lines and lines from points. Then we look at the example and may be we are wiser then. So, what is currently considered a line or a point can be flexible, which is exactly the point (khem) of the non-euclidean geometries.

# Algebra

Algebra is a devil's gift to a mathematician to forfeit geometry.

Expressing some mathematical issue with numbers, or with variables and their relations, and not with points and lines, helps. Well, it steals the geometric visual perception from you, but gives an important and easy-to-use set of tools.

Algebra as such developed later than geometry, but! It's not like people who did geometry in Ancient Greece or Egypt did not know about numbers. So, while the math of a Hellenistic civilisation would be more focused on geometry than algebra, it's not that they could not count or something.

# Computer science

Well, there is not computer science without computers, but informatics or cybernetics deal with information processing. And even humans are computers. The (mentioned above) Antikythera mechanism is a computer.

So, even if the raw computational power might be a limit, information processing as such does not need silicon chips.

# History

(Or: people from older days are not dumb.) To put more things into perspective:

• Logarithms became widespread thanks to Napier and Co. in the 16th-17th century. They were known to Indian mathematicians already in 2 century BC!
• Cardano is said to develop first concepts around complex numbers in 1545.

If you want a society that knows mostly geometry, then you are wrong in 18th century. You need Hellas around Euclid or Pythagoras. And yes, even they (as noted in other answers) knew a lot about other fields of mathematics.

If you aim something like 18th century, they know enough of mathematics to stray astray from the pure geometry path.

# Development path

Generally, mathematics expands into every direction it deems usable. You might want to use some historic issues to steer this movement into your alternative universe, or just pop in some mathematical jewels that are a bit off-side from the today's generic path. So, I'm just throwing ideas in.

• Don't kill Evariste Galois early.
• Earlier of more wide adoption of logarithms as an export from India, together with their number system (that we know as Arabic numbers).
• Continued fractions are a nice and beautiful mathematical theory, useful for things like design of gears or approximating the astronomical year with a proper calendar. (There are some ideas that might be better suitable than Gregorian calendar, for example. This alone might give your alternative universe a very distinctive and memorable characteristic.)
• Look into the fate of Cardano, for example.
• Finally, just don't kill off Archimedes early. Because, well,

Don't touch my circles!

Creating a computer doesn't require numbers; it requires a discrete representation.

# Turing Machines

The theoretical machine that stands alongside Lambda Calculus as the beginning of all things computer science actually doesn't need a single number to make sense. What you actually need here is sets and graphs. Sets describe the possible symbols that can be read from and written to an infinite tape. This shouldn't be a terribly foreign concept to anyone with a writing system. Graphs describe possible states of the machine and the logic of how symbols should be written and how the tape should be moved around. Once again, this shouldn't be a foreign concept to a society that routinely draws things with a straightedge and compass to make "calculations", one that presumably has a way of explaining algorithms.

## The issue: computation is discrete while geometry is continuous

If all you have is geometry, your computations really aren't going to do you any good without some way to numerically measure shapes. With only a discrete representation, you basically have to invent numbers to create any sort of useful bridge between geometry and computers. You can still do all sorts of fancy non-numeric computation such as string manipulation, so I still wouldn't call this useless.

In the end, without numbers, you'll have to make special purpose mechanical devices that help to measure ratios such as golden ratio calipers. These will be far more useful than any computer if you have no concept of numbers.

### Nitpicking: not having numbers has some interesting implications

Finance and inventory are nonsensical without at least having counting numbers. A shepherd will want to know how many sheep he has and will want a way to record that. A merchant will need to know how much gold to expect from a transaction. You simply can't have trade (and thus society) without numbers. It's unlikely that a society would develop advanced geometry but not numbers because even though the two concepts are mostly orthogonal, numbers are much more useful to survival and societal cooperation.

• If a guard puts a bean in a bag at every enterer, and removes one at every exiter, he can know when the building's empty again. In a second bag, add but don't remove any, and you have a variable holding today's count, without doing any actual counting or knowing how to express the number in language. If tomorrow's bag is heavier than today's bag, visitors went up. Change bags every hour, you can graph visitor trends visually. Automate these things and I believe there's a lot of useful computing that be done without having a number system. Mar 21, 2019 at 4:01
• The MONIAC has numbers on its gauges, but couldn't a variant of the beans-bag method be used instead? A society could build an economic calculator without having numbers. en.wikipedia.org/wiki/MONIAC Mar 21, 2019 at 4:07