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The Greek philosopher Empedocles famously postulated that the four elements comprising all matter are Fire (F), Water (W), Earth (E), and Air (A). More recently, the video game “Doodle God” has turned this theory into a fun way to waste time: by repeatedly combining these $4$ elements, the player can construct novel entities. For example:

Dust = Earth + Air = EA
Energy = Air + Fire = AF
Storm = Energy + Air = (AF)A
Swamp = Earth + Water = EW
Life = Energy + Swamp = (AF)(EW)

Here are some of the more complex advanced entities that can be created:

Quicksilver = W(F(A(FE)))
Glass = F(A(A(FE)))
Lizard = ((A(FE))((AF)(EW)))(EW)

Note that parentheses must be used to disambiguate these combinations, because “combination” is a non-associative operation, i.e. A(BC) is not necessarily the same as (AB)C.

QUESTION: Can we come up with a Doodle-God-esque theory of matter that starts with fewer starting elements? Can we come up with one that uses only one starting element to generate all other entities, but still uses roughly intuitive/guessable rules of combination? How would you generate Fire, Water, Earth, and Air from your proposed “first element” and what would the intermediate steps be?

(Here’s an analogous concept in SKI combinator calculus, for you computer science nerds.)

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    $\begingroup$ Ever heard of taoism? $\endgroup$
    – The Square-Cube Law
    Jun 30, 2020 at 4:18
  • $\begingroup$ The Peano axiomatic construction of the natural numbers famously uses only one primitive element, namely zero, and one rule of construction, namely the successor function. So that one is by definition the successor of zero, two is the successor of the successor of zero, three is the successor of the successor or the successor of zero and so on. And once you have the naturals you can construct the integers, and then the rationals, and then the reals and finally the complex numbers. All from one primitive element and one primitive constructor. $\endgroup$
    – AlexP
    Jun 30, 2020 at 7:04
  • $\begingroup$ Sounds like rubik cube ;P $\endgroup$
    – user6760
    Jun 30, 2020 at 10:28
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    $\begingroup$ @AlexP a Peano-like construction would give you essentially a cookie-clicker-esque game: just keep clicking on your element and get the next element. You need a binary operator, like Doodle God has, to make things somewhat interesting. $\endgroup$
    – Aetol
    Jun 30, 2020 at 12:59
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    $\begingroup$ @Aetol: And addition, subtraction, multiplication, division, exponentiation, and logarithm are what? The point was that the entire arithmetic and theory of numbers can be constructed from one primitive element and one primitive unary operation. If one primitive element and one primitive unary operation are enough to construct the entire arithmetic and theory of numbers then they are most certainly enough to construct a video game; after all, that game is implemented as a computer program, which is an application of arithmetic... $\endgroup$
    – AlexP
    Jun 30, 2020 at 14:19

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Yes.

Your basic element is the empty set.

Combination forms a new set with two elements: the operands of the combination.

At the lowest level, I think you will have to be content to discard intuition, but you can easily non-intuitively assign meanings to four (or however many you like) simple constructs, and let those drive the intuition for more complex combinations, a la Doodle God.

Fire, water, earth, and air could, for example, be as follows: Air: {} (the empty set) Fire: {{}{}} (Air+Air) Water: {{{}{}}{}} (Fire+Air) Earth: {{{}{}}{{}{}}} (Fire+Fire)

But it need not be. That's just the four simplest constructions I could think of.

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    $\begingroup$ You can make this more compact if you allow order to be meaningful (Use lists rather than sets). Then Water can be: [[[][]][]] (Fire+Air) and Earth: [[][[][]]] (Air+Fire) $\endgroup$
    – Joe Bloggs
    Jul 1, 2020 at 14:01
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Yes and No

Yes, we can start with fewer elements - as few as two.

No, we can't start with one. Let's say that 4 units of the basic element (A) create compound B, and one B and two As creates C. You have twelve units of A. Does it create 3 Bs, or 2 Cs? Even assuming non-association, strict additive compounds make it pretty much impossible to make a chemistry system.

There's a reason why the calculi you linked (and their associated languages) have two symbols - that is the minimum syntax required to express anything. Let's let our two elements be Light and Darkness.

Let's let the synthesis of Light and Darkness be Aether.

So Light = 1, Darkness = 0, and Aether = (10) or (01).

We then can build Ephemera (Aether + Light), and Tactile (Aether + Dark).

And then Fire (Ephemera + Dark), Air (Ephemera + Light), Water (Tactile + Light) and Earth (Tactile + Dark).

And, having constructed those four elements, you can then build anything you could previously build with the four elements.

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    $\begingroup$ Actually, iota and jot are separate languages. Iota has only one combinator, which is in fact sufficient (because parentheses and non-associativity allow for greater variety). Your “impossibility proof” for one element doesn’t work. But I like your light and darkness system! (+1) $\endgroup$
    – Franklin Pezzuti Dyer
    Jun 30, 2020 at 1:04
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    $\begingroup$ Let's say that 4 units of the basic element (A) create compound B, and one B and two As creates C. You have twelve units of A. Does it create 3 Bs, or 2 Cs? (AAAA)(AAAA)(AAAA) creates 3 Bs. ((AAAA)AA)((AAAA)AA) creates 2 Cs. You can see this in real-life chemistry, where adding reactants in different orders can (sometimes) result in different products. $\endgroup$
    – cowlinator
    Jun 30, 2020 at 1:11
  • $\begingroup$ That's only true because you have a standard unit (the iota, in that case). Two iotas becomes (ii), which is a discrete unit. But how much (base element) is one base element, and how much is two? If people are combining arbitrary quantities of an element, you can't have stoichiometry with only one starting ingredient. There's the possibility of allotropic configurations, but it still seems impossible to come up with a "chemistry" with consistent behaviour. $\endgroup$
    – jdunlop
    Jun 30, 2020 at 1:20
  • $\begingroup$ So if you have a cup full of A, do you actually have a cup full of A, or is it a cup full of B? Can you ever have A? $\endgroup$
    – jdunlop
    Jun 30, 2020 at 1:22
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    $\begingroup$ This answer appears to assume that this chemical system needs to be able to be manipulated by a person. This need not be the case. Constructing chemicals in this universe could be a matter similar to fusion, where you can't precisely control the products you get. With that allowance, yes, you could get by with one base item. $\endgroup$
    – Rithaniel
    Jun 30, 2020 at 5:16
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One could describe the actual system as having 3 meta-elements.

Or 3 subatomic particles: the proton, neutron and electron. Rearrangements of these 3 are responsible for all the elements, and further combinations of elements comprise all matter.

You could have such a system with 2 elements with a redefinition. For the composition of elements, the presence of a proton implies the presence of the electron and for purposes of matter, they could be considered a package.

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    $\begingroup$ Note that a description of neutronless baryonic matter requires two different combination operators, though--nuclear combination, and chemical combination. $\endgroup$
    – Logan R. Kearsley
    Jun 30, 2020 at 15:39
  • $\begingroup$ @LoganR.Kearsley - since parenthesis are allowed, put the nuclear combos in parenthesis and then combine those groups chemically.. Or brackets! There could be brackets. Also ellipses. <And these things>. Yeah, all set. $\endgroup$
    – Willk
    Jun 30, 2020 at 18:21
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Basically, what you are describing here is an algebraic "magma" or "groupoid." You have a set of things (elements) that you can combine with each other, via a single operation, to attain other things in that set. In this context, your question can be interpreted as "Can a magma be generated by a single item?" Very directly speaking: Yes, absolutely.

To illustrate this, consider the elements of our reality. What makes a particular element be a particular element? The number of protons in the nucleus. Change the number of neutrons and you get an isotope. Change the number of electrons and you get an ion. Properties might change, but it's still the same element. Now, suppose our "operation" were "when combining two elements, just add the number of protons together." Hydrogen has one proton, so hydrogen mixed with hydrogen has two, and gives helium. Helium and helium would be beryllium. Etcetera etcetera.

This kind of system would be, in one sense, more well behaved than the Doodle God system, as it would be associative. However, if the combination is automatic, it would also be poorly behaved because, if you just put a billion hydrogen atoms together, instead of getting a gas, you would get a single atom of whatever element you assign to have a billion protons. So, you probably would want to have some sort of trigger on it, such as the combination requiring a large amount of pressure. (Nuclear fusion, anyone?)

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Yes, you can have just one element.

According to Greek philosophy, everything is made of a base element called Prima Materia (well, actually it's technically Arche, but the term Prima Materia is used more often these days).

The ultimate Unobtainium, Prima Materia is described as being pure potentiality of being. It holds the potential to become Anything and Everything; however, it remains formless when left to itself. In order for it to become something like lead, gold*, or a sheep, it must be acted upon by a εἶδος (Form). These "Forms" are ideas of such overwhelming potency that they warp reality, bringing order to the chaos of Prima Materia.

* Incidentally, this mutability was the basis of Alchemy; the purpose of the Philosopher's Stone was to turn matter back into Prima Materia, so they could then turn it into gold.

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The Universe is Just Energy All the Way Down

In science,the Theory of Everything is the hypothetical that the 4 fundamental forces of the universe (electromagnetism, gravity, strong force, and weak force) are all caused by a single underlying force. Although the exact solution for theory of everything has not been unequivocally proven, one model that seems relevant to your question is the theory of Quantum Gravity.

QG theory stipulates that the entire universe is a matrix of binary energy states that are either occupied or not occupied at the planck scale, and based on what pattern of occupied states you have determines the manifestation of various subatomic particles.

So, if your Doodle God game were to start with just the element "energy", this element could be combined with itself to form the false vacuum of space. Then adding more energy to the false vacuum would create your fundamental subatomic particles such as electrons, quarks, and gluons. Then you could combine quarks and gluons to get protons and neutrons. combine protons and neutrons to get atomic nuclei, then combine atomic nuclei with electrons to get hydrogen. Hydrogen + hydrogen gets you stars and helium... and well if you see where I am going with this you just keep building up your stars until you get various kinds of star deaths that result in the elements of the periodic table. Once you have these you can go to town making planets, organic molecules, microbial life, and well... you've played Doodle God, so you know how it will expand outward from here.

In this version of doodle god, fire, air, water, and earth would be mid-game elements rather than fundamental ones.

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