01 December 2018
Created: 2018-12-01
Updated: 2019-06-23
Topics: Mathematics, Physics, Philosophy, Cognitive Science
Confidence: Speculative
Status: Still in progress
The following is a distillation of an argument found in this Nautilus Article: How Chaos Makes the Multiverse Unnecessary by Noson S. Yanofsky, plus my accompanying thoughts…
1) Is it possible to describe the entire universe in the language of mathematics?
2) We can postulate that the universe actually doesn’t have much structure in it.
3) The reason we believe the universe is structured is due to the fact that a scientist acts like a sieve and focuses only on those phenomena that are predictable.
“It is almost a tautology: Science predicts predictable phenomena”
4) Scientists stick to studying only those phenomena that possess symmetry, since if there are symmetries then there exists conservation laws and constants (roughly Noether’s Theorem). We see structure because we are only selecting a subset of phenomena to study, namely those that have structure.
— brief digression into maths —
5) We learn about the Real numbers in school and how we can add, subtract, multiply, etc. The Real #s have an important property that they are ordered (we can always tell one is less than another).
6) Later we learn about the Complex #s, which are created by adding together two parts: a real # plus another term which is the product of a real # and an imaginary #… ( complex # = r1 + r2*i ) However, unlike the real #s the complex #s are not totally ordered (and if we do order them, the ordering will not respect the multiplication of complex numbers).
7) The fact that the complex #s are not totally ordered means that we lose structure when we go from the real to the complex #s.
8) Just as you can construct the complex numbers from pairs of real numbers, so too can you construct another type of number called the quaternions from pairs of complex numbers ( quaternion # = c1 + c2j …. which can be rearranged into quaternion # = r1 + r2i + r3j + r4k, where i,j,k are special numbers similar to how previously we had an imaginary number i )
9) We can think of the complex #s as two-dimensional, with the quaternions as four-dimensional, possessing the complex #s as two-dimensional subsets of it.
10) Like the complex #s, the quaternions also fail to be ordered AND also lose another property called commutativity ( q1q2 does not necessarily equal q2q1 ). As such, the quaternions lack even less structure.
11) We can keep doing this doubling of our number systems (called the Cayley-Dickson construction). Given a certain type of number system, one gets another system that is twice the dimension of the original system, but also has less structure (fewer axioms or properties) than the starting system.
12) Doubling again, we can get the Octonions, from which we now also lose the associative property: o1(o2o3) does not necessarily equal (o1o2)o3. We can keep doing this to get Sedonions and so on, but let’s stick with the Octonions for now.
— back to science —
13) The real #s are used in all aspects of science. Quantities, measurements, and lengths are given in real #s. Later on, physicists started needing complex #s to describe waves, quantum mechanics, etc. As of late, quaternions are now trying to be used for some phenomena.
14) We usually think of the real #s as most fundamental while the other larger number systems are not. However, rather than look at the real #s as being most fundamental, let’s flip this around and think of the Octonions as being fundamental and the other smaller number systems as just being special subsets of octonions (or really subgroups, subfields, etc).
15) Because real #s have more structure and we lose some of that structure as we go to the complex #s and so on, by looking at the flipped perspective if we wanted a particular system that obeyed particular rules or properties of our choosing, all we have to do is look at a particular subset of the Octonions. In other words, if a system has a particular structure, special subsets of that system will have more structure, or more properties / satisfy more axioms than the starting system.
16) The above is analogous to what we do in science. We do not look at all phenomena, but rather study only those subsets of phenomena that satisfy the requirements or properties of symmetry and predictability…I’m looking at you Karl Popper! ;)
17) Notice that the math used for a subset is easier for us than the math used for the bigger sets. This is because in those subsets we have more axioms or properties or assumptions available for us to work with. As such, a subset of our phenomena around us is easier to describe with a law of nature stated in math. In contrast, when we look at all phenomena, like our Universe, it is harder to find a law of nature described by math.
— moving onwards —
18) As physics advances and our goals shift to wanting to describe larger and larger phenomena and systems, larger classes of mathematical systems are needed at the cost of fewer and fewer axioms, and hence lesser and lesser structure. Is the universe in its totality really structure-less?
19) Following this trend, with respect to the desire to describe every part of our whole universe, is the structure we see dependent upon which particular properties we desire at our scale and system of perspective? (like for instance our desire for the properties of symmetry and predictability?)
— and as always, my thoughts are —
20) Does our hope for finding Truth and understanding Reality lie in finding or creating other types of Minds other than our own, with their own desires, scales of perspective, and differences in what they find inherently intuitive or aesthetically pleasing? …different minds as different sieves of reality.
As such, your pick: aliens or robots?
Alternatively, is the future of science as a human endeavor starting to become outdated? (whether from the perspective of the aforementioned argument, the diminishing returns of science, or even why most published research findings are false) Will the future of science all be done by machines with only various levels of interpreters for us mere mortals?
Or should we design an entirely new language for the purposes of truth-seeking, like some Sapir-Whorfian and E.O. Wilson’s Consilience concept mashup? Something that is maybe like part deductive logico-maths, part aesthetics like music or visual imagery, or even some non-phonologically-sequential based language in our heads (Ted Chiang / Arrival style, where our inner voice radically changes in form)? Is the future of Science just pure Creativity & Art, stretching the limits of how we can express ourselves and what is even possible to express?
Or will Science take on a whole new process or methodology, like a Preferential Attachment or Yule Process type of concept found in market systems or Evolution itself (ie - evolutionary algorithms)? What if the future of Science was just all about sex, mutation, reproduction, breeding new organisms, and spreading / permuting Life amongst the stars and that that is how we “discover Truth”…namely by what type of forms, shapes, functions, biochemistries, anatomies, organisms, and ecosystems develop and are conserved in various environments?