Back to Tegmark’s strange paper on the mathematical universe, as promised earlier. I will now attack his main thesis, rather than some obscure bit in the middle. Tegmark defines two hypotheses:
The External Reality Hypothesis (ERH) states that there exists an external reality completely independent of humans.
The Mathematical Universe Hypothesis (MUH) states that our external physical reality is a mathematical structure.
Tegmark’s main thesis is that the ERH implies the MUH. The ERH is evidently something that people intuitively favour. Scientists, especially, like to think that humans are mere observers of and participants in, not creators, of physical reality. But the MUH is strongly counter-intuitive. Our world seems very material, not abstract. So claiming that the ERH implies the MUH is a radical violation of our intuitions.
To get from the ERH to the MUH, Tegmark uses the concept of “baggage”. He doesn’t explicitly define “baggage” anywhere. He does, however, leave clues to what it is, such as the following:
The ERH implies that for a description to be complete, it must be well-defined also according to non-human sentient entities (say aliens or future supercomputers) that lack the common understanding of concepts that we humans have evolved, e.g. “particle”, “observation” or indeed any other English words. Put differently, such a description must be expressible in a form that is devoid of human “baggage”.
In this passage, he seems to imply that baggage is simply the characteristics of reality that are dependent on concepts special to humans.
Immediately after, however, he shifts to a rather different notion of baggage. He lays out a hierarchical tree of our theories about the world, ranging from quantum mechanics to sociology. He says of these theories:
All these theories have two components: mathematical equations and ‘baggage’, words that explain how they are connected to what we humans observe and intuitively understand… At each new level in the hierarchy of theories, new concepts (e.g. protons, atoms, cells, organisms, cultures) are introduced because they are convenient, capturing the essence of what is going on without recourse to the more fundamental theory above it. It is important to remember, however, that it is we humans who introduce these concepts and words for them: in principle, everything could have been derived from the fundamental theory at the top of the tree, although such an extreme reductionist approach appears useless in practice. Crudely speaking, the ratio of equations to baggage decreases as we move down the tree, dropping near zero for highly applied fields such as medicine and sociology.
I would say he is speaking not just crudely, but erroneously. It is true that as we move away from more mathematical theories towards the so-called softer disciplines, the language used shades more into everyday intuitive concepts. However, that does not imply that the concepts and entities in the softer disciplines are any less real from a possible extraterrestial being perspective than they are from a human perspective. The so-called fuzzy terminology of the soft disciplines could still capture boundaries and modules in reality that exist independently of the perspective sentient beings bring to them. Lack of clarity does not imply lack of objectivity. His earlier definition of baggage as the characteristics of reality that depend on concepts particular to humans is unrelated to his later, implied definition of baggage as anything that is described in non-mathematical human language.
This is fatal to his inference from the ERH to the MUH. For he reasons as follows:
1. The ERH implies that a “theory of everything” has no baggage.
2. Something that has a baggage-free description is precisely a mathematical structure.
Taken togther, these imply that the ERH implies the MUH. However, as I pointed out, the ERH implies a theory of everything has no baggage, in the sense of having no concepts that are particular to a human perspective. This, however, is not the sense of “baggage” being used in 2. Something can have a baggage-free description, in the meaning of baggage used in 1., without being a mathematical structure. We could imagine that nature has certain clearly delineated boundaries and modules, which humans see fit to describe in increasingly “fuzzy” and intuitive terms as they move from basic physical theories into more complex theories of reality. But since these boundaries and modules are real, aliens would also tend to describe them the same way. The fuzziness and intuitiveness of the language humans use in no way implies that they are talking about less objective entities when they use that kind of language.