Dear Dr. Mitra,
I have enjoyed reading your paper on a mathematical universe without
postulates, and thank you for referencing my recent writing on the
subject. I hope you appreciate me taking the time to discuss with you
on a few of the finer points below. Naturally, there is more to say
when it comes to points of disagreement than of agreement, so for
brevity I have mainly focused on my points of disagreement.
"One has to assume a measure over the set of all possible mathematical
pmodels, anthropic reasoning and observations can then be used to
compute probabilities of being in some universe. But this isn't
fundamentally different from how cosmologists actually attempt to
explain the observable universe we find ourselves in."
I would disagree with the assessment that this isn't fundamentally
different in that I wouldn't say cosmologists generally attempt to
explain the observable "universe" we find ourselves in at all; rather,
they attempt to explain the observable surroundings within our
universe (and yes, they use anthropic reasoning for that).
The fundamental difference is merely in the limitation of how far this
anthropic reasoning can be applied. While cosmologists are quite
comfortable applying anthropic reasoning to select one's planet, solar
system, galaxy, etc...and perhaps even to select one's 'universe'
within 'the multiverse' (if one ascribes to such a theory) -- the
difference is that in that case, multiple universes are predicted to
exist on the basis of a single mathematical model that is assumed to
be "the one true model of reality."
In contrast, the view proposed by Tegmark and myself is that there is
no "one true" mathematical model. The "universe" of a theory should
refer to all things describable by the mathematics of that theory.
Then, given the viewpoint that the choice of mathematics is arbitrary,
one may apply anthropic reasoning to explain why one observes certain
mathematical principles about the laws of physics.
As an aside, I would like to clarify one point that I did not make
clear in my paper, but that which we as a community should be careful
with when discussing such topics. Specifically, when discussing
theories such as my own or Tegmarks which postulate the existence of
infinite universes, we should be careful not refer to "the probability
of being in some universe" because this is technically zero for all
universes. To be more precise, one should instead refer to "the
probability that an observer shall observe some particular property of
his universe to hold" because this requires an integration over the
multidimensional probability density of universes.
"What I will argue for in this essay is something radically different.
Like Tegmark, I will argue for a mathematical universe, but each
element of this multiverse is an observer, not some universe. By
doing that I make sure that no physical baggage gets smuggled in via
the backdoor."
This does not strike me as fundamentally different from what was
proposed by either Tegmark or myself. An observer must be some
"element" as you say that is describable by the mathematical laws of
the universe in which it exists. Because the universe is described by
mathematical theorems, this implies that everything which exists in
the universe is a manifestation (real or imagined) of some set of
mathematical statements that describe the thing in relation to the
axioms. Fundamentally, any such thing may be equally considered as an
"observer" but it does not make sense to consider a non-conscious
"viewpoint" which is why Tegmark only bothered to refer to SAS as
observers -- and the same in my paper, although I did not use the term
SAS. This creates a dichotomy between "universes that contain
self-awareness" and "universes that do not contain
self-awareness"...in other words, universes that are perceived as
being physical realities from some perspective, vs those that are not
perceived as being real by anyone.
"..here the laws of physics are meta-laws that describe the
mathematical multiverse. These laws should in principle follow from
mathematics alone, so there isn't any room for postulates at the
fundamental level" and in your conclusion, "If indeed there only
exists a mathematical multiverse, the known laws of physics should
follow from pure mathematics without any assumptions."
I like your description of the laws of physics as meta-laws, because
the laws of physics are formulated from our local viewpoint, based on
human-scale phenomena, as a set of simplified rules for explaining
reality. In other words, they tend to describe the behavior of
complex systems consisting of a nearly infinite number of smaller
individual parts that behave with consistent local rules, and thereby
allow the system as a whole to be modeled with a more simple
analytically approachable equation (for example, fluid dynamics or
maxwells equations).
However, in light of this, I do not feel that your quest to show that
such equations can be derived from mathematics alone is
justified...because they are, by nature, merely approximations to the
"truer" equations that describe the constituent particles in those
complex systems. Of course, this is somewhat semantical, as we may
say that we are striving for a less approximated set of laws of
physics -- and in my paper, when I say "laws of physics" I am always
referring to a set of precise mathematical rules, as opposed to the
approximations from complex systems.
It is interesting that you say "...should follow from pure mathematics
without any assumptions" because mathematics is not at all free of
assumptions. Mathematics is merely the set of formal language, and
every mathematical concept is relative to some set of axioms, which
are by definition assumptions.
Nonetheless, I certainly recognize the importance of seeking to
explain the universe while being free from postulates...because until
we stop making arbitrary assumptions, we cannot escape the logical
infinite regress of explanations. An infinite regress is no
explanation at all, and the presence of our existence is demonstrable
proof that some explanation must exist; therefore, we must seek an
explanation without postulates.
However, as I proved in section 2.3 of my paper (and separately, in
bottom-up fashion in section 2.2), it is necessarily NOT possible to
derive our laws of physics (neither in exact form, nor approximate)
while being free from postulates, which further proves that there can
be no one-true axiomatic system that is taken as "the" laws of
reality/physics...which further proves that our notion of existence
must necessarily be relative and not absolute; in other words, the
notion of "physical reality" is, as you alluded to initially, an
illusion. Thus, I actually feel that (contrary to your concluion) it
IS clear how to describe the laws of physics while being free from
postulates. It is precisely what I did in my paper, and it merely
requires accepting that existence is relative -- which itself need not
be taken on faith, because it was proven by contradiction.
Regardless of when we may disagree or agree, I certainly think we have
more in common than most when it comes to thinking about such issues,
and I look forward to further scholarly discussions with you, directly
or indirectly.
Cheers,
Stuart B. Heinrich
