A World Without Mathematics

What significant technologies would be lost in a world without "Advanced" Mathematics? Specifically, what technologies required to get to the next level of scientific progress in an area would be lost. Your largest constraint is algebra. The humans on this world can count to small numbers, they can perform addition, subtraction, comparison, and other mathematics operations below the level of multiplication and addition on whole integers, you may consider variables as part of our working algebra set and zero may or may not exist.

I realize some basic Chemistry might be possible and that tinkering with ratios might lead to division since a ratio can be expressed that way, yet it may also be ignored like the number zero in our own history. So unless something explicitly requires more advanced math it is not removed as a possibility. Modern Chemistry, however, would definitely be removed as we play with algebra too much.

Geometry and Logic as areas of Mathematics have special consideration. Logic can be formulated without advanced algebra, although its level of utility and formalization may depend on it. Geometry is constrained to matching shapes and lengths as well as other things that are capable of being eyeballed. Anything beyond that should be kept to a minimum.

As a good test to constrain your technologies, ask if processes can be done via a "match" unit versus a unit requiring conversion. If you can directly perform comparisons it can be done with minimal math. Conversion of units however requires multiplication or division and is not allowed.

• Another way of thinking of the question: "Given an army of very smart eight-year olds and a few caveman grunts how much technology could be mustered?" – Black Oct 22 '14 at 3:05
• I propose the "very smart eight year olds" comment as the new title :-) – Mourdos Oct 22 '14 at 9:00
• IIRC there are still societies on Earth that lack mathematics beyond basic counting -- as in, "one, two, three, lots" -- which have basic agriculture. You lose the ability to use money without higher numbers, although you can use it without algebra. – Blazemonger Oct 22 '14 at 13:38
• @Blazemonger But even so, they still have Sesame Street. – Shokhet Oct 22 '14 at 13:47
• I presume you mean "Algebraic manipulation" instead of "Algebra". Eight-year-olds discussing homomorphisms and Galois theory seems to me somehow unnatural. – imallett Oct 22 '14 at 17:41

Essentially you end up with very little, maths makes the world go around.

Without maths you have no:

• Computers
• Anything that requires computers
• Precision engineering and manufacturing
• Physics
• Elements of chemistry
• Economics
• Elements of biology
• Astronomy (in terms of understanding orbits, etc)
• And the list goes on, and on, and on.

Now, some things can be replaced by iterative trial and error. For example once you work out that gunpowder goes boom you can experiment with ratios to find the strongest mix. There would be no way to calculate the ratios of gears in an engine but again you could keep trying different combinations until you got what you needed.

There would most likely be books published full of nothing but various tables, similar to the old logarithm tables. These tables would allow you to look up your designed inputs and outputs and it would then tell you the combination of gears to use.

What I don't understand though is why the leap to abstraction would not be made. Once you've generated the table of gear ratios the patterns within them should be identifiable. Once you've identified the pattern you essentially have algebra, since all algebra is is a way to describe that pattern.

• That last paragraph is a very good point, essentially the opposite of the question -- 'at what level of technology would it become unbelievable for people not to develop advanced mathematics?' – evilsoup Oct 22 '14 at 9:27
• The point you make about a leap to abstraction doesn't really stand up because of the prerequisite of good reading and writing skills. Without such skills they couldn't write down tables of gear ratios etc. I made a (quick and incomplete) edit to my original post to avoid a too large comment. – Epsilon Oct 22 '14 at 13:48
• @NickR The question said nothing about reading or writing skills either way. Lack of mathematics doesn't automatically imply lack of reading/writing. – Tim B Oct 22 '14 at 13:57
• @TimB Is it credible that advanced reading and writing skills could be developed by people whose mathematics could not develop beyond an 8 year old level. Indeed, number and counting seem to be more innate conceptually than reading and writing. We certainly counted (mentally) before we developed language. As an aside : Looking at your profile pic, I see Watford has changed a lot since I left London. (bad joke). – Epsilon Oct 22 '14 at 14:03
• @NickR It's possible, not probable but possible. All I'm saying is that you're making assumptions. For example if this is an alien species we're talking about they might have excellent reading/writing but struggle with abstract concepts. – Tim B Oct 22 '14 at 14:25

I'll answer this question based on the clarifying comment given describing "an army of smart eight year olds".

While I have no doubt that a very clever eight-year old can perform demanding mathematical calculations, I have serious doubts that such a child could achieve the necessary conceptual understanding to formulate such calculations in the first place.

For example, little Mr. Smarty Pants may be able to write down something like: $$\frac{d x^2}{dx} = 2 \cdot x$$ if they were correctly taught the method of differentiating a polynomial. However, it defies credulity that he could conceptualize and formalize the necessary underlying mathematics to pose the question in the first place. Such methods require considerable abstraction and generalization.

Therefore, one has to assume that "an army of smart eight-year olds" would be limited to elementary aspects of counting, arithmetic, geometry, and perhaps elementary aspects of linear algebra. In the absence of a conceptual framework, these sorts of skills would have limited value in developing technologies.

In essence, I believe that such mathematical knowledge could only serve to modestly enhance naive technologies, by which I mean those technologies not requiring mathematical knowledge.

There are today isolated tribes of people whose mathematical knowledge has never surpassed this level. Their technologies are very modest. Fire. Basic materials technologies - mostly for weapons and shelter. Some basic agricultural knowledge. And of course combinations of these technologies, such as cooking.

EDIT Regarding comments made elsewhere concerning abstraction, take the skills of reading and writing. There are today many 3 and 4 year old children who can read and write to an elementary level. However, such skills are dependent on the teaching skills developed by adults and those skills are dependent on abstraction and generalization of a very sophisticated type and covering a wide range of subject. Consider how long it took for literacy to become commonplace and how it demands post-8-year-old levels of knowledge.

• I see you grasped what I was getting at with the comment. If I'm reading you right, your placing a loose upper limit on technology level at that of the isolated tribes that exist today. Correct? – Black Oct 22 '14 at 7:08
• I would not underestimate the effects of experience and experimentation. Such a society may not understand the mathematical concepts behind technology they're using, but that does not mean they can not figure out how it works solely by experimentation. Still, it would be very primitive. Just not completly stone age. – Scorpio Oct 22 '14 at 7:20
• @Black Perhaps a limit on the type of technology rather than strict application. Experience would allow for the continuous refinement of existing technologies, at least as far as an individual technology permits. There would be an upper bound on the refinements, but that upper bound need not ever by met (in practice). For example, crop irrigation would be achievable within the limits of particular techniques but not necessarily gross area of irrigation, implying that large populations could be supported. The particulars of the environment would be critical. Nice OP question! – Epsilon Oct 22 '14 at 13:28
• @Scorpio I agree. However, the level of an 8 year-old is a serious restriction. Comprehension, as well as reading and writing skills (i.e., the ability to pass on previously obtained knowledge) would seriously impair revolutionary developments. If one posits cumulative knowledge building, then one quickly looses sight of the 8 year-old abilities. Even Wilma Flintstone had beautiful clothes and nifty dish washer! – Epsilon Oct 22 '14 at 13:32

This is interesting because what mathematics tells us is how things work, and consequently allows us to predict other things that will work or extend solutions to be more efficient, which is how we have made a lot of our progress for the last three hundred years or so.

So you might start by saying that without advanced mathematics you could get to a level of technology approximately commensurate with what we achieved by the mid seventeenth century, which was when people like Newton and Leibniz were building the foundations of much of the mathematics that we use on a day to day basis.

But consider some problems that are very mathematically complex and have been challenging to reproduce- if I throw you a ball and you catch it, you're doing a lot of processing ( from a mathematical perspective ) very rapidly. Likewise every time you recognise a face you are doing something that is exceedingly hard to replicate through mathematical means.

It certainly doesn't seem impossible that one could have a species that could innately do things that we need technology for without needing to understand why they work. Indeed this is one of the concepts in Stephen King's Tommyknockers - the aliens don't know ( or care ) why their technology works, they are just naturally able to construct it, which also has the benefit of making them very alien to us.

• Interesting take on the question. The separation of action and explanation is intriguing. I can see no fundamental need to know how something works if you can just do it, interestingly our typical conception of magic fits this view perfectly. While this does answer in the spirit of the question I don't know if you could justify calling anything you do this way a technology. – Black Oct 23 '14 at 0:14
• @Black any sufficiently advanced magic is indistinguishable from technology. ;p – glenatron Oct 23 '14 at 16:24
• The ability to naturally construct things without the need for understanding them is something that seems to me like a very viable evolutionary pathway along the lines of our ability to anticipate the motion of an object in flight. Perhaps they see things in nature and can mimic their behaviour ( so aircraft develop based on flying creatures and so on ) without the why being necessary, like some birds can mimic speech without understanding it. This would just be a natural aptitude. – glenatron Oct 23 '14 at 16:34

A very disorganized world, and a communist one at that.

Without mathematics, or any attempt to understand mathematics, any invention will be a creation of pure guesswork, and very difficult to reproduce accurately. Complicated technology would be possible - harvesting, irrigation, wheels, telescopics, very crude electricty, steam power - but without a way to represent how these advances were achieved with any accuracy, reproductions will be crude and inefficient, and improvement almost impossible.

Economics would take the hardest hit. With no way to measure the value of goods and services, trade would be on-the-fly and individuals would often end up with much less or much more than they needed, leading to excess for some and poverty for many. Societies would have to gather all their resources into a single communal place for any equitable exchange to occur.

You would get something like an Iriqoui tribe, which makes sense - they didn't have much use for math. Focus would be placed on improving the natural bounty of the world, without a heavy focus on measuring how much improvement has been done or on efficiency in methods. Things would get accomplished, but with much trial-and-error. And most of the best ways to do things would be passed down through stories rather than formulas.

In short, you wouldn't get very many empires (though you would get some as DVK points out, they would be limited in size, scale and type), you would get a lot of communistic sharing in small tribes, and you might get some remarkable advances in agriculture, but technological advances would stagnate due to their irreplicability.

• "you wouldn't get very many empires" - I'm unsure that Hittites had advanced algebraic knowledge. Actually I'm almost sure they didn't. – user4239 Oct 22 '14 at 15:31
• @DVK True, and a few ancient societies didn't rely nearly as much on mathematics to succeed. – Zibbobz Oct 22 '14 at 15:42
• Good point on the economy! It hadn't even occurred to me how hard it would be hit. – Black Oct 22 '14 at 23:56

To illustrate what would happen without maths, you could look at how Newton came about his laws of motion.

The path was from high quality observations made by Tycho Brahe, Keppler's laws and Newton's concept of force (I simplify, I know).

Tycho Brahe could easily make these observations without advanced maths. Keppler could not have discovered that the vector from the center of motion to the planet swept an equal area over an equal amount of time without somewhat advanced maths. Newton needed to build on calculus to formulate his laws.

Though it was in general understood that the earth moved around the sun well before this, to understand why would, in my opinion, be impossible.

EDIT: But where would the development in maths have stopped? I think the best place to break the development we have known is the concept of functions. Without functions calculus would not exist. Why functions would never be thought off is a bit tricky to come up with a good reason.

• According to Wikipedia, the notion that the Earth revolves around the Sun had been proposed as early as the 3rd century BC by Aristarchus of Samos. Writing doesn't require algebra, nor does basic astronomy; simply spend a night sky gazing and notice that perhaps the Earth moves instead of the sky lights. – Cees Timmerman Oct 22 '14 at 13:45

This is a tricky question because the lines you are drawing are kind of arbitrary. Multiplication and division are just fancy adding if you really get down to the nuts and bolts.

For the question as written you would be limited to pretty basic construction. No aqueducts, only the most basic of pumps, multistory buildings would be major undertakings. Trade would be rudimentary, a single currency would be possible but once you start looking at trade accross borders you need conversions and things fall back to straight barter (although whether barter would even work without knowing "2 cows = 5 goats" is arguable with the question as written).

• While your points are good and quite valid, I might add that your thinking too mathematically. One does not need to know that they're getting an equal deal of cows for their goats. They just need to be satisfied with it. Your points still stand though. – Black Oct 23 '14 at 0:04
• Consider what the Romans managed to do. Aqueducts, mines, land surveying, pretty fancy logistics and bridges. And of course different currencies and impressive trade networks. – Walter A. Aprile Oct 23 '14 at 15:07
• @WalterA.Aprile Romans studied geometry and used the abacus. They had surpassed the level of mathematical knowledge prohibitted in the question as written. – Myles Oct 23 '14 at 18:36

Looking at our own history, there is a logical answer. However, you would have to give the society multiplication and division but I may be able to convince you to do that!

The four operations of addition, subtraction, multiplication, and division are all ancient operations that were independently discovered on numerous occasions. My oldest daughter was four years old when she "discovered" multiplication: If daddy, mommy, and her each have two ears then how many ears in total in the family?

Now look at our own mathematics to see where we were stuck until a genius came along to advance us. I would say that the first true "stuck" point was before calculus: that would have been Newton and Leibniz who got us out of that one. A world without Newton would be pretty much stuck at addition, subtraction, multiplication, and division. Everything before Newton is rather natural and intuitive, and there would be no convincing argument as to why multiplication and division did not develop.

You could "stick" the society at pre-Newton levels of mathematics and see where they go. The beginnings of the industrial revolution would likely have happened, including early textile machines such as the Flying Shuttle and the Spinning Jenny. Likewise we would have likely seem some advancement in iron-making as well. However complex machines would be right out. I would put the absolute limit at Carnot's work: controlled combustion and thermodynamic understanding would be completely impossible.

• Actually before the enlightment there was another boom in mathematics± the ancient era (greeks etc) and pythagoras, hypocrates, euclides. Those spawned many basic ideas such as geometry - however looking from a point of view BEFORE that time those things are far from trivial to proof. As such it can be called "advanced" too. Without this we would basically be stuck in a pre-roman era, where the maximum span between pilars is 3 meters.. – paul23 Oct 23 '14 at 12:28
• Many of the ancient advances are considered trivial and elementary. If Pythagoras had not been around to give us the length of the hypotenuse, somebody else would have. Though both Leibniz and Newton worked on calculus, we could have potentially waited millennia before somebody else stumbled across it. – dotancohen Oct 23 '14 at 12:42
• Same can be said for newton (if it wasn't Newton, Maxwell did similar things 100 years later), and or Einstein (if it wasn't for Einstein we'd have niels bohr or ). It's all a product of social cultures. – paul23 Oct 23 '14 at 14:22
• I don't know about Maxwell's ability to invent calculus, honestly I know too little about him other than his experiments. But Bohr replacing Einstein: no way. Bohr did important work, but I absolutely refuse to believe that he would have come up with GR or SR. – dotancohen Oct 23 '14 at 14:27
• Maxwell is the one who made the maxwell equations - mathematically proving how magnetic and electric are linked, he is to faraday/ampere as newton was to kepler/galileo. - But even while netwon was busy there were others - leibniz is just as likely considered a father of calculus, it's just that newton -being from the UK- is credited. And what about Fourier? That's just what I'm trying to say: a civilization without calculus but the same culture WILL create calculus sooner rather than later. – paul23 Oct 23 '14 at 21:15

Without multiplication you can't do a lot of modern commerce. You can't sell 244 units each for 2.45$if you can't calculate 244 * 2.45. Niall Ferguson argues in "The Ascend of Money" that new math notation was crucial for having the ability to make loans for precise amounts of interest and then use the money of those loans to build businesses. Without those businesses you get problems with a lot of technology development. • I can buy the second part, but the first is absolutely false. Selling 244 units each for$2.45 each would only require you to do 244 transactions to avoid multiplication. You could even see how many units fit in a particular container, sum the price of each object as you remove it and set a price for a case. Not to say that it doesn't hinder commerce to remove quicker math such as multiplication. – Black Oct 26 '14 at 16:12
• @Black : Doing 244 transactions instead of doing 1 transaction produces a significant burden on commerce. That burden matters. Time is a highly useful resource. – Christian Oct 26 '14 at 16:18
• Agreed. "Not to say that it doesn't hinder commerce...". Pointless from the view of this question, but interesting, is how much this would hinder commerce. Off the top of my head I can't think of any mass traded objects Medieval Europe and prior. – Black Oct 26 '14 at 16:33
• @Black That's the point. In Medieval Europe they didn't had arabic numbers that allow for easy multiplication. – Christian Oct 26 '14 at 16:40

• Somebody could operate an abacus without really understanding what’s going on. Of course, in our world, there was somebody who knew what was going on and built the abacus, but it is also conceivable that somebody could have painstakingly experimented with moving beans between bowls and empirically arrived at the conclusion that certain operations yield useful results. An extreme example would be a computer: It can perform complicated mathematical operations without understanding any of them.

• Kepler’s laws (see also Bent Nielsen’s answer). They are complicated laws describing empirical geometrical observations but miss the dynamical aspects to an extent that is almost unrelatable for people used to our modern view on the problem. Still, it is conceivable that we would have continued with this approach to celestial mechanics (though it would have taken us much longer).

• As the example of economy was given: If you do not have large numbers, you can just stack different coinage levels (think of the British pre-decimal coinage taken to extremes). Somebody could have devised a complicated system for comparing magnitudes using weights which are appended to a scale at different levers (I am pretty certain, such a thing actually existed).

• There exists complicated mechanical devices for drawing certain mathematical figures or estimating areas and curve lengths etc.

• Even today, we find new ways to apply mathematics to long-existing concepts gaining new insights and people (usually from non-mathematical fields) reinvent mathematics all the time – e.g., this paper reinvented numerical integration.

This leaves you at the following problems:

• Where do you draw the line to mathematics? At the end of the day, mathematics is “only” the abstraction of things that already exist in reality. You can do a lot of mathematics without doing mathematics openly or even noticing that you are doing it. I think you can get surprisingly far without ever writing down an equation or having an elaborated concept of numbers (though it will be much more difficult, of course).
• Some mathematical relationships beg to be discovered under given circumstances. If they do not get deduced in a formalised mathematical way, another way will turn up which hides the mathematics – even to its inventor, if necessary. Mathematics does not need to be proven, it can be explored empirically (though this is arguably the more difficult way).
• What makes your world to be without mathematics? Is there a cultural prohibition? Are your people just unable to abstract in certain ways? Is everybody just stupid? This question may be tho most important, because it decides to what extent your people can circumvent mathematics.

Nothing would be affected by the removal of mathematics. Since everything can boil down to binary, boolean logic can replace mathematics. Mathematics is simply pure logic, and removal of high-level mathematics is simply removal of high-level thinking of logic.

You can relate this to high-level programming languages and low level programming languages.

High-level programming languages, such as C/C++/Java/etc allow us to communicate and work with the low level logic, such as assembly or machine language. In this case, the highlevel languages are proportional to mathematics, and assembly/machine languages/etc are pure logic.

Rollercoaster Tycoon was written mostly in Assembly, so it goes to show that even pure logic can generate advanced technologies.

"You can represent any algorithm, or any electronic computer circuit, using a system of boolean equations."

• While it may be true that everything can reduce to 0s and 1s, and it may not be. It still begs the question of working with such a system and why it would not be considered Mathematics in base 2 (opposed to base 10). Certain operations would be just as hard to learn in that system. Getting to the core of the question, how do you multiply in base 2? Can you logic your way into an algorithm? It's obviously possible but illustrates how much of your childhood you take for granted. Math was not easy and we've been bootstrapped into higher math by time and teachers. – Black Aug 19 '15 at 3:30