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Non binary computers, in particular ternary computers, have been built in the past (emphasis mine).

One early calculating machine, built by Thomas Fowler entirely from wood in 1840, operated in balanced ternary. The first modern, electronic ternary computer Setun was built in 1958 in the Soviet Union at the Moscow State University by Nikolay Brusentsov, and it had notable advantages over the binary computers which eventually replaced it, such as lower electricity consumption and lower production cost.

If you want to make ternary computer the standards, I think you should leverage on those advantages: make energy more expensive, so that saving energy is a big advantage, and make production more expensive.

Note that, since smelting silicon is an energy intensive activity, already increasing the cost of energy will indirectly affect the production costs.

Instead of avoiding it, transcend binary:

Either let the evolution of technology take its course and somehow create a demand for non-binary processors. Analogous to what is happening now in the crypto currency scene: The developers of IOTA based their project on a ternary architecture model and are even working on a ternary processor (JINN).

Or let aggressive patenting and licensing in the early stages of binary processors (e.g. a general patent for binary processors due to lobbying or misjudgements in the patent office) be the cause for starting work on non-binary processors with less restrictive and more collaborative patents.

I would like to advance the idea of an analog computer.

Analog computers are something like the holy grail of electronics. They have the potential of nearly infinite more computing power, limited only by the voltgae or current measuring discriminator (i.e., the precision of measuring an electric state or condition).

The reason we don't have them is because using transistors in their switching mode is simple. Simple, simple, simple. So simple, that defaulting everything to the lowest common denominator (binary, single-variable logic) was obvious.

But even today, change is coming.

Analog computing, which was the predominant form of high-performance computing well into the 1970s, has largely been forgotten since today's stored program digital computers took over. But the time is ripe to change this. (Source)

If analog and hybrid computers were so valuable half a century ago, why did they disappear, leaving almost no trace? The reasons had to do with the limitations of 1970s technology: Essentially, they were too hard to design, build, operate, and maintain. But analog computers and digital-analog hybrids built with today’s technology wouldn’t suffer the same shortcomings, which is why significant work is now going on in analog computing in the context of machine learning, machine intelligence, and biomimetic circuits.

...

They were complex, quirky machines, requiring specially trained personnel to understand and run them—a fact that played a role in their demise.

Another factor in their downfall was that by the 1960s digital computers were making large strides, thanks to their many advantages: straightforward programmability, algorithmic operation, ease of storage, high precision, and an ability to handle problems of any size, given enough time. (Source)

But, how to get there without getting hung up on the digital world?

• A breakthrough in discrimination. Transistors, for all their value, are only as good as their manufacturing process. The more precisely constructed the transistor, the more precise the voltage measurement can be. The more precise the voltage measurement, the greater the programatic value of a change in voltage = faster computing and (best of all for most space applications) faster reaction to the environment.




• Breakthrough in modeling equations. Digital computers are, by comparison, trivial to program (hence, BASIC). Their inefficiency is irrelevant compared to their ease of use. However, this is because double-integration is a whomping difficult thing to do on paper, much less to describe such that a machine can process it. But, what if we could have languages like Wolfram, R, or Haskell without having to go through the digital revolution of BASIC, PASCAL, FORTRAN, and C first? Our view of programming is very much based on how we perceive (or are influenced by) the nature of computation. Had someone come up with an efficient and flexible mathematical language before the discovery of switching transistors... the world would have changed forever.

Would this entirely remove digital from the picture?

Heck, no. That's like saying the development of a practical Lamborghini (if the word practical can ever be applied to a Lamborghini) before, say, the Edsel would mean we would have never seen the Datsun B210. The single biggest weakness of analog computing is the human-to-machine interface. The ability to compute in real time rather than through a series of discrete often barely related steps is how our brains work — but that doesn't translate well to telling a machine how to do its job. The odds are good that a hybrid machine (digital interface to an analog core) would be the final solution (as it will today). Is this germain to your question? Not particularly.

Conclusion

Two breakthroughs: one in transistor manufacture and the other in symbolic programming, are all that would be needed to advance analog computation with all of its limitless computational power over digital computing.

Having thought about this and looked at L.Dutch's answer, I may withdraw my original answer (or leave it just for interest).

Instead I will give a political answer.

As mentioned by L.Dutch, the Soviets came up with a ternary system (see below). Because of the limited use of the Russian language throughout the world the Soviets often resented the fact that US scientific papers got more credence - after all English is the Lingua Franca of science. (This is true by the way, not a fiction, I'll look for references).

Suppose the Russians had won a war over the West. It was common in Soviet Russia for science to be heavily politicised (again I'll look for references). Therefore, regardless of the validity of a non-binary system the Russians could have mandated ternary or some other base simply as a form of triumphalism.

Note - I'm chickening out of finding references at the moment. I've found some but they involve delving into Marxist doctrine or buying an expensive book. My personal knowledge of the situation came from talking to a British scientist who was digging through old Russian papers looking for bits that had been missed or had been distorted by doctrine. Maybe I'll delve further but not right now.

The first modern, electronic ternary computer Setun was built in 1958
in the Soviet Union at the Moscow State University by Nikolay
Brusentsov

https://en.wikipedia.org/wiki/Ternary_computer

Get rid of George Boole, inventor of Boolean Algebra, probably the main mathematical foundation of computer logic.

Without Boolean Algebra, regular algebra would give quite an edge to decimal computers, even if you needed three to four times as much hardware per digit.

There's no need to kill him, just have something happen that stops his research or get him interested in another field instead.

EDIT - On reading the answer by L.Dutch, I see that there is an energy-saving argument for using trinary. I'd be interested to find out how theoretically true that is. Crucially the OP talks about transistors rather than thermionic valves and that could make a difference. There are also other energy questions to address other than the simple switching of a transistor. It would be good to know the extent of this saving and any extra cost associated with building and maintaining the hardware. Heat dissipation may also be an issue.

I remain open-minded as well as interested in this approach.

I don't think there is a historical justification for your premise as far as transistors are concerned so instead, I will just say:

The minimum historical change is No Electronics

It's possible to use other bases but just a really bad idea.

IBM 1620 Model I, Level H

IBM 1620 data processing machine with IBM 1627 plotter, on display at
the 1962 Seattle World's Fair The IBM 1620 was announced by IBM on
October 21, 1959,[1] and marketed as an inexpensive "scientific
computer".[2] After a total production of about two thousand machines,
it was withdrawn on November 19, 1970. Modified versions of the 1620
were used as the CPU of the IBM 1710 and IBM 1720 Industrial Process
Control Systems (making it the first digital computer considered
reliable enough for real-time process control of factory
equipment)[citation needed].

Being variable word length decimal, as opposed to
fixed-word-length pure binary, made it an especially attractive first
computer to learn on – and hundreds of thousands of students had their
first experiences with a computer on the IBM 1620.

https://en.wikipedia.org/wiki/IBM_1620

The key phrase there is variable word length decimal which is a real faff and actually still uses binary at the electronic level.

Reasoning

Any other electronic system than binary will soon evolve into binary because it depends on digital electronics.

It is commonly supposed, by those not in the know, that zero voltage represent a binary zero and some arbitrary voltage, e.g. 5 volts, represents a 1. However in the real world these voltages are never so precise. It is much easier to have two ranges with a specified changeover point.

Having to maintain say ten different voltages for ten different digits would be incredibly expensive to make, unreliable and not worth the effort.

So your minimum historical change is No Electronics.

As I understand it, early tribes used base 12 and it's a lot more flexible than 10--they had a way to count to 12 by counting knuckles to get up to 60 on two hands pretty easily which is the basis of our "Degrees".

10-finger-counters supposedly defeated the base 12ers but kept their time system and degree-based trigonometry.

If the base 12ers had won, a three-state computer might have made a LOT more sense (Binary might have actually looked silly). In this case A byte would probably be 8 tri-state bits (let's call it 8/3) which would perfectly fit 2 base-12 digits instead of our 8/2 layout which always had a bit of a mis-match.

We tried to cope with our mismatch by using BCD and throwing away 6 states from each nibble (1/2 byte) for a more close approximation of base 10 which gave us a "Pure" math without all these weird binary oddities you get (like how in base 10, 1 byte holds 256 states, 2 bytes hold 65536, etc)

With 3/8, base 12ers would have no mismatch, it would be really clean. Round 3-bit numbers would often look like nice base12 numbers: 1 byte would hold 100 states, and 2 bytes would hold 10000, etc.

So can you change the numeric base of your book? Shouldn't come up too often :) It would be fun to even number pages in base 12... complete submersion.

The strength of binary is that it's fundamentally a yes/no logic system, the weakness of binary is that it is fundamentally a yes/no logic system, you need multiple layers of logic to create "yes and" statements with binary logic. The smallest change you would need to make to change away from binary (in terms of having the rest of the world being the same but computing being different) would be to have the people who pioneered the science of computers, particularly Turing (thanks @Renan) aim for, and demand, more complex arrays of basic logic outcomes (a, b, c, etc... vary combinations, all of the above, none of the above). Complex outcome options require more complex inputs, more complex logic gates and a more complex programming language: consequently computers will be more expensive, more delicate, and harder to program.

A few people might mess around with binary for really basic machines, like pocket calculators, but true computers will be more complex machines.

Decimal computers.

Modern computers are, indeed, binary. Binary is the classification of an electrical signal as occupying one of two states, conditional on the voltage. For the sake of simplicity, you could say that in a 5V system, anything signal above 4V is a '1' and everything else is a '0'. Once a signal has been confined to two states, it's pretty easy to apply Boolean math, which was already well-explored ahead of computers. Binary was an easy choice for computers because so much work was already done in the area of Boolean algebra.

When we needed to increase the range of numbers, we added more signals. Two signals (two bits) could represent 4 distinct values. 3 bits - 8 values, and so-on. But what if, instead of adding more signals to expand our values, we simply divided the existing signals up more. In a 5V system, one signal could represent a number from 1-10 if we divide up the voltage. 0-0.25 volts = 0. 0.25-0.50 volts = 1. 0.50-0.75 volts = 2, etc. In theory, each signal would carry 5x the data a binary signal could. But why stop there? Why not split each signal into 100 distinct values?

Well, for the same reason we never went further than binary - environmental interference and lack of precision components. You need to be able to precisely measure the voltages to determine the value, and if those voltages change, your system becomes unreliable. All types of factors can affect electrical voltages, RF, temperature, humidity, metal density, etc. As components age, their tolerances tend to decrease.

Any number of things could have changed this - if you use a different medium - light, for example, interference isn't a concern. This is exactly why fiber-optics can carry so much more data than electrical connections.

The discovery of a room-temperature superconductor could also have allowed different computers to become standard. A superconductor doesn't lose electrons to heat. This means you could pump more voltage through a system without fear of overheating, requiring less precise components and less (no) cooling.

So, in-short, binary computers dominate because of physical limitations related to electricity and the the wealth of knowledge (Boolean Algebra) that was already available when vacuum tubes, transistors, and semiconductors came about. Change any of those factors, and binary computers may never have been.

If there were a technology for maintaining single atoms of hydrogen in an excited state indefinitely; destructive readout by letting it decay and detecting the photon. So logic base could be however many orbitals you can control.

In the late 1950s analog computers were developed using a hydraulic technology called fluidics. Fluidic processing is still used in automatic transmissions, although newer designs are hybrid electronic/fluidic systems.