The Mathematical Multiverse and the Meta-Measure Problem
- Elie Feder
- Nov 21, 2024
- 18 min read
Updated: Mar 9
If you thought multiverse scientists have difficulty explaining the fine tuning of the constants without God, you won’t believe the trouble they have explaining away the designed laws of nature. The mathematical multiverse solution to the design of the laws is so outlandish it makes a standard multiverse look tame by comparison. It’s really the ultimate multiverse theory - and it runs directly into the ultimate meta-measure problem.
Highlights of this essay:
Below is an essay version of the ideas presented in Episode 11 of Season 2 of the Physics to God podcast. You can hear the audio version above.
Introduction
In our first series about an intelligent cause, we presented three independent arguments from modern physics for the existence of God: in essays 1-5 from the fine tuning of the constants, in essay 7 from the design of the qualitative laws of nature, and in essays 8-9 from the ordering of the initial conditions.
This essay will address how some physicists use the multiverse as an alternative to the second argument for God, from the design of the qualitative laws of nature. Before doing so, let’s recall the main idea of the argument. It began with the mystery that perplexed physicists: Our universe is governed by general relativity and quantum mechanics. But why do these laws of nature exist instead of some other laws?
The solution to this mystery emerged from the recognition that the laws of our universe are not any random set of laws but are rather perfectly designed to allow for the emergence of our complex universe. Their exquisite design indicates that their cause is an intelligent designer who specifically selected them for the purpose of bringing about our complex universe.
Until now, we’ve lumped together the way the multiverse explains the designed laws with the way it explains the fine tuned constants and the ordered initial conditions. However, things aren’t so simple. Trying to explain our special laws by making recourse to an infinite varied multiverse, with each universe having its own laws, has problems above and beyond those faced by other multiverse theories - it runs into a more advanced form of the measure problem - the meta-measure problem.
As we discussed in our first series, many scientists don’t even try to use a multiverse to explain the designed laws of nature. This is because, in contrast to the seemingly arbitrary constants and initial conditions, the qualitative laws are relatively simple, beautiful, and symmetrical. Therefore, many scientists are content to ignore their apparent design and simply posit that these laws just exist because they do. As such, they don’t feel the same need to posit a multiverse to explain them.
We, and some scientists, disagree with this approach because it’s arbitrary to posit that only these laws are real when we know that many other laws are logically and physically possible. We think an intelligent cause is a better solution than just accepting the existence of a specific set of special laws as an unexplained brute fact of reality.
Physicist Max Tegmark, like us, is bothered why these laws exist as opposed to any others. But in contrast to us, he invokes a multiverse to explain their apparent design. This essay will focus on his attempt.
The Mathematical Universe Hypothesis
While the idea of a multiverse has been informally discussed as far back as Greek philosophers, in 2003 Max Tegmark proposed a scientific categorization of the different levels of the multiverse.
Here’s his formulation of the four levels:
(1) A multiverse with different initial conditions in each universe;
(2) A multiverse with different constants of nature in each universe;
(3) The Many Worlds Interpretation of quantum mechanics;
(4) The Mathematical Universe Hypothesis.
Though we haven’t explicitly mentioned Tegmark’s four levels, we’ve already discussed the first three. Today we’re going to discuss his level four multiverse - what he calls the Mathematical Universe Hypothesis or the “Ultimate Ensemble.” This fourth level addresses the problem of explaining our seemingly special laws of nature without an intelligent designer.
Tegmark’s Mathematical Universe Hypothesis emerges as a solution to the basic question that we just mentioned regarding the qualitative laws of our universe. Since there are infinitely many possible sets of laws, why should our universe operate under this set of laws and not any other mathematically consistent set of laws?
Tegmark summarizes his approach in his 2007 paper titled The Mathematical Universe, as follows:
If the Theory of Everything…exists and is one day discovered, then an embarrassing question remains, as emphasized by John Archibald Wheeler: Why these particular equations, not others? Could there really be a fundamental, unexplained ontological asymmetry built into the very heart of reality, splitting mathematical structures into two classes, those with and without physical existence? After all, a mathematical structure is not “created” and doesn’t exist “somewhere”. It just exists. As a way out of this philosophical conundrum, I have suggested that complete mathematical democracy holds: that mathematical existence and physical existence are equivalent, so that all mathematical structures have the same ontological status.
The Mathematical Universe Hypothesis rests upon two distinct ideas. The first is the hypothesis that mathematical existence is equivalent to physical existence. Essentially, this assertion claims that not only can mathematics be used to describe the physical world but that mathematics and physics are essentially identical, being two sides of the same coin.
Since, according to Tegmark’s first idea, mathematical existence is equivalent to physical existence, and in the realm of mathematics, all mathematical structures are equally valid, his second idea naturally follows. Namely, every possible set of mathematical laws has a real physical expression and governs an independently existing universe.
If you accept these two premises, then according to the Mathematical Universe Hypothesis, not only does the multiverse have universes with different initial conditions and different constants of nature, but it also has a separate universe for each possible mathematical law of nature. For instance, some universes are governed by general relativity and quantum mechanics while others are not; some have three dimensions of space while others have 754 dimensions; some are governed by ordinary laws of nature while others are governed by mathematically consistent magical spells, and so on. The possibilities are literally endless.
The Level 4 multiverse is so expansive that Tegmark argues there can’t even be a fifth level multiverse. This is because the Mathematical Universe Hypothesis already includes every possible multiverse theory that can be formulated mathematically. This is why Tegmark calls this multiverse the Ultimate Ensemble.
To simplify matters and to emphasize the centrality of mathematics in Tegmark’s level 4 multiverse, we’ll call it the mathematical multiverse.
While Tegmark calls it the Mathematical Universe Hypothesis or the Ultimate Ensemble, we're not calling it the mathematical multiverse just to be different or to give a plug for mathematics. It’s actually important that this multiverse is based on ascribing physical reality to mathematics.
After positing the mathematical multiverse, Tegmark explains the apparent design in our laws just as multiverse theorists explain our fine tuned constants. He argues that all possible mathematically consistent laws actually exist, with the vast majority being “poorly designed.” However, the reason we don’t observe those other laws is simply because we could only exist in a universe that’s governed by our apparently well-designed laws. As such, Tegmark argues that a mathematical multiverse fully explains our special laws without any need for an intelligent designer.
Evidence for the Mathematical Multiverse
Tegmark is making a really big claim by equating math with physics and positing that every possible mathematical equation governs a real physical universe. You may be wondering what kind of evidence he brings to support this incredibly bold assertion.
He starts with the question of why these laws exist as opposed to any other laws. Of course, a possible answer to this question is that an intelligent cause intentionally selected these laws for the purpose of bringing about a complex structured universe. But, just like many other scientists, Tegmark harbors the unjustified premise that God doesn’t exist (as we discussed in essay four). Therefore, Tegmark argues that the only way to avoid the problem of why our particular laws of nature are real is to posit that all possible laws are real. In other words, the only way to explain the one set of designed laws that we observe is to posit a mathematical multiverse that contains every possible set of laws.
Tegmark posits “complete mathematical democracy” to avoid an asymmetry in the fundamental constituents of nature whereby one set of initial conditions, one set of values for the constants, and one set of mathematical laws happen to exist, as opposed to all the other infinite possibilities.
Put simply, Tegmark considers the fact that our observed laws actually exist as opposed to any others, to be such a compelling problem that it simply cannot be true. Instead, every possible set of laws must also exist! Therein lies the totality of his support for the real existence of a mathematical multiverse.
It’s amazing how once you believe God is impossible, the only solution left is to posit that every single possible equation really exists. It just shows you why you need God to ground science. We’ll come back to this point later.
Are Simple Laws of Nature Typical?
For the moment, let’s grant Tegmark his two premises that physical existence is equivalent to mathematical existence, and that every possible set of mathematical laws has a real physical expression in its own independently existing universe. Working with these assumptions, let’s examine whether the mathematical multiverse is capable of establishing the Typical Universe Premise.
It's been a while since we discussed the Typical Universe Premise, so let's review its meaning and importance. As we explained in essays 6 and 7, the typical universe premise maintains that from the set of all universes in the multiverse containing intelligent observers, ours is typical. Applied to Tegmark’s theory, the mathematical multiverse must maintain and justify that our laws of nature are a typical set of laws that produce intelligent observers. This premise is necessary in order to invoke chance to explain our laws while simultaneously avoiding the problems of a naive multiverse that can explain anything and everything.
While this may sound simple, it’s a very dubious claim indeed. In fact, the exact opposite seems to be true - our simple laws seem quite special. To the best of our knowledge, the first to clearly formulate this problem with the Mathematical Universe Hypothesis was physicist Alexander Vilenkin in his 2006 book Many Worlds in One: The Search for Other Universes (page 203). He wrote as follows:
The number of mathematical structures increases with increasing complexity, suggesting that 'typical' structures should be horrendously large and cumbersome. This seems to be in conflict with the beauty and simplicity of the theories describing our world.
Vilenkin’s critique of the mathematical multiverse is that the laws in our universe are very simple and elegant. While it’s hard to rigorously determine what speculated universes with other laws of nature would even look like, it would intuitively seem that the number of complex and awkward laws that are consistent with intelligent observers is much larger than the number of simple and beautiful ones.
This is because there are many more ways to formulate complex equations than simple equations, and there doesn’t seem to be any intrinsic connection between intelligent observers and simple laws. Therefore, a typical universe in the mathematical multiverse would be expected to have complex laws, not simple laws like our own.
This is similar to essay 8’s Boltzmann Brain and Grand Universe problems - problems that emerged in the context of a regular multiverse. Just as a regular multiverse seems to falsely predict that the typical observer in the typical universe is a single brain surrounded by chaos (as opposed to real observers like us), so too the mathematical multiverse seems to falsely predict that the typical intelligent observer will exist in a universe with extremely complex laws (as opposed to our observed simple laws).
A false prediction is obviously a major problem. After all, if the mathematical multiverse fails to establish the Typical Universe Premise and reverts to the claim that in an infinite varied multiverse everything has to happen somewhere, it becomes a naive multiverse which is a theory of the gaps that can explain anything and everything, and therefore explains nothing at all. Tegmark, like all other multiverse theorists, must find a way to make our universe typical or he will be unable to solve the very problem he set out to answer - namely, why do we observe these laws and not any others?
Another Designed Measure
Tegmark attempts to establish the Typical Universe Premise in a similar manner as multiverse scientists who defend eternal inflation. Namely, he introduces a measure that states that in a Level 4 multiverse, a universe is more likely to have simple laws than complex laws. If this measure governs probabilities in the multiverse, then our universe with simple laws will be typical.
Tegmark writes as follows:
If you thought that was bad, consider the problem of assigning statistical weights to different mathematical structures at Level IV. The fact that our universe seems relatively simple has led many people to suggest that the correct measure somehow involves complexity…If there is such an exponential penalty for high complexity, we should probably expect to find ourselves inhabiting one of the simplest mathematical structures complex enough to contain observers.
While the approach of positing a measure that makes simple laws more likely than complex laws may seem promising to Tegmark, it’s subject to the same line of questioning that plagued eternal inflation theorists. That is, what selected this particular measure from the set of all possible measures? Instead of the multiverse having a measure that makes it more likely to have simple laws, why doesn’t it have a measure that makes it more likely to have complex laws?
By introducing this measure, Tegmark has merely substituted design in the laws of nature with design in the measure that chooses a probability distribution for the laws of nature. This doesn’t explain what caused our laws to be simple, elegant, and apparently designed, but merely pushes the problem back to explaining what caused the mathematical multiverse’s measure to be designed.
The only way that a measure can support the Typical Universe Premise and thereby render our universe typical is if the measure itself is designed. In other words, even assuming the existence of Tegmark’s measure, the fact that the mathematical multiverse has just the right measure to result in its typical universe having simple, beautiful laws like our own directly points to an intelligent designer who specifically chose this measure from all others.
Meta-Measure Problem
Unfortunately for Tegmark, the problem gets even worse. Even if he arbitrarily posits a “simple-law measure”, the mathematical multiverse still can’t establish the Typical Universe Premise. To appreciate why, remember what originally motivated Tegmark’s theory - the desire to avoid an asymmetry in the laws of nature. (Specifically, why do quantum mechanics and general relativity exist as real laws, as opposed to all other possible laws?)
But at the end of the day, once Tegmark introduces one particular measure that chooses the laws of nature in the mathematical multiverse, he’s merely replacing an asymmetry in the laws of nature with an asymmetry in the measure of nature which acts upon the laws.
Vilenkin expressed the problem as follows:
I am not sure that the notion of weights for the set of all mathematical structures is even logically consistent: it seems to introduce an additional mathematical structure, but all of them are supposed to be already included in the set.
The problem is, that according to Tegmark’s logic, the ultimate ensemble multiverse includes every possible multiverse theory that can be formulated mathematically. That means that just as there’s a “simple-law measure” that makes universes with simple laws more likely, there’s also a “complex-law measure” that makes universes with complex laws more likely. Again, since the entire basis of Tegmark’s theory is to avoid a mathematical asymmetry in reality, he must logically infer that every possible measure – including both the simple-law measure and the complex-law measure - must actually exist as part of the Level 4 multiverse. Following Tegmark’s logic, how can it be that one measure is more real than all the others?
But if all measures are equally real, it’s baseless to argue that a typical universe in the mathematical multiverse has simple laws because the mathematical multiverse has a simple-law measure; it would be equally logical to suggest that the typical universe has complex laws because another realm of the vast Level 4 multiverse has a complex-law measure.
If Tegmark would attempt to avoid this problem and cling to the measure approach to establish the Typical Universe Premise, he would need to introduce a meta-measure; this meta-measure would act upon regular measures to ensure that there are more multiverses with simple-law measures than with complex-law measures.
Of course, we would reply just as above: according to this reasoning, it doesn’t make sense to have an asymmetry in meta-measures. Therefore, there must also exist other meta-measures that ensure there are more multiverses with complex-law measures than with simple-law measures, and all those meta-measures are also part of the Level 4 multiverse, and so on.
Following this line of reasoning ad infinitum reveals that Tegmark is stuck in an infinite regress. Since every possible measure (or meta-measure) is itself a mathematical law, it must already be included in the Ultimate Ensemble as a description of a different universe in the Level 4 multiverse.
In summation, in attempting to explain our apparently designed qualitative laws of nature without an intelligent designer, Tegmark’s theory posits a mathematical multiverse, an Ultimate Ensemble that contains all possible mathematical laws. Besides confronting the standard measure problem and its implied fine tuning and design, this theory faces a unique problem in attempting to use measures to justify the Typical Universe Premise. Namely, if the mathematical multiverse truly contains all possible mathematical laws, then it’s inconsistent to say that there’s only one measure that determines which laws are typical. Thus, the attempt to use a mathematical multiverse to explain the design of the laws of nature can’t use an extrinsic measure to establish the Typical Universe Premise, and therefore it amounts to nothing more than a naive multiverse of the gaps.
The Difference between Science and Mathematics
Even though we’ve shown that the mathematical multiverse inevitably falls prey to the meta-measure problem, remember that we only got there based on the assumption that the mathematical multiverse’s two premises were correct. It’s worthwhile to analyze those conceptual premises that underlie the theory to see if they were ever likely to be true in the first place.
The first premise was the complete identification of mathematics with physics. From this followed the second premise: just like all equations have mathematical existence, so too, they all have real physical existence.
There’s a superficial basis for Tegmark’s first claim that math is the same as physics. If you open any advanced modern physics textbook, it’s hardly distinguishable from a math textbook. It’s almost entirely written in mathematical symbols. This might lead you to believe that Tegmark is truly onto something and that it’s reasonable to conclude that the two studies are, in fact, identical.
But this observation is only surface-deep. One way to see this is by noticing that while physics books do have lots of mathematics, most math books have absolutely no physics. On a basic level, this should clue you into the fact that they’re very different subjects. Let’s analyze this difference on a deeper level.
Science is the study of the actual physical universe that we observe with our senses. Specifically, physics is the branch of science that studies the different forms of energy within spacetime and the laws that govern their interactions. Given that science deals with our actual universe, scientific theories are derived from direct empirical observations together with inferences from those observations. Likewise, scientific theories are verified through experimentation and comparison to the actual physical universe.
Mathematics, on the other hand, is the abstract study of number and quantity; a study that we can only access through our minds. Since mathematical objects exist outside the framework of space and time, mathematical theorems aren’t subject to observation and experimentation and are only verified by assessing the validity and consistency of their logical deductions.
Some philosophers argue that mathematical objects only exist in the human mind, in which case the contrast between physical objects and mathematical objects is even more glaring. But even Mathematical Platonists, who maintain that mathematical objects have their own realm of existence independent of our minds, still recognize that the realm of existence of mathematical objects is qualitatively different from that of physical objects. This is because, in contrast to physical objects, mathematical objects don’t exist in spacetime - meaning, even if they exist on some plane of reality, it’s not the reality addressed by physics.
Don’t worry if you don’t follow this last point. The key idea is that physical objects are known through observation in space and time, while mathematical concepts are only known through pure thought; and scientific theories are verified through experiments and observations, while mathematical theorems are verified through logical deductions from first principles. Math and science are clearly two different subjects.
In light of this distinction between math and science, let’s consider Tegmark’s second claim: that just like in mathematics it doesn’t make sense to say that one equation is more real than another, so too in physics it doesn’t make sense to say that any one set of physical laws is more real than any other.
In abstract mathematics, Tegmark is correct – there can’t be an asymmetry whereby one equation is more truly “mathematical” than any other. This is because mathematical equations don’t get their reality from being observed or tested in the physical world. Rather, mathematics is an a priori abstract study that grants an equal claim on mathematical truth to all consistent mathematical structures that logically follow from a system’s axioms or first principles.
However, science is based on a fundamental asymmetry: the one physical world we observe is real and the imaginary worlds we dream up are make-believe. On an even deeper level, were it not for the fact that we observe one physical reality, we would never even know that a physical world exists! That is why scientists must first observe how nature actually behaves and only afterward attempt to find the right laws that govern the particular reality they’ve observed. You can’t eliminate the essential role of observation in science by simply positing the truth of every possible law you dream up.
The Relationship between Science and Mathematics
While Tegmark’s equation of physics and mathematics does a disservice to both, we don’t want to create the impression that they have nothing to do with each other. Despite the significant difference between science and mathematics, there’s a deep and amazing connection between them: mathematics is the special language used to describe the laws of the physical universe.
Galileo Galilei, one of the fathers of modern science, was the first to state this relationship clearly. In listening to the following quote from Galileo, note that when he says “philosophy,” he’s referring to what we would call “science” today.
Galileo wrote as follows:
Philosophy is written in that great book which ever lies before our eyes — I mean the universe — but we cannot understand it if we do not first learn the language and grasp the symbols in which it is written. This book is written in the mathematical language, and the symbols are triangles, circles and other geometrical figures, without whose help it is impossible to comprehend a single word of it; without which one wanders in vain through a dark labyrinth.
Albert Einstein succinctly stated the difference between math and physics, as well as the significant relationship between them, as follows:
Physics is essentially an intuitive and concrete science. Mathematics is only a means for expressing the laws that govern phenomena.
The reason for this deep relationship between mathematics and physics lies in math being a unique language particularly suited for describing the physical world in a clear and rigorous manner. In the words of Richard Feynman:
Mathematics is not just another language. Mathematics is a language plus reasoning; it is like a language plus logic. Mathematics is a tool for reasoning. It is in fact a big collection of the results of some person’s careful thought and reasoning. By mathematics it is possible to connect one statement to another. – Richard Feynman, The Character of Physical Law, pg.40
Using the tool of mathematics, scientists infer quantitative conclusions that should result from their hypothesized theories and then empirically test whether these predictions correspond with actual physical observations. Math’s ability to enable experimental testing makes it the perfect language for formulating and verifying scientific theories.
We can now understand the textbooks. Every modern physics textbook is filled with mathematical equations because math is the indispensable language for formulating the laws of physics. But most math books have no physics because math itself has no essential connection to the physical universe.
Why Science Needs God
Let’s now tie the various ideas of this essay together. Everyone can understand why there’s no asymmetry in mathematics. All consistent mathematical equations are equally real. But the same can’t be said about the laws of physics. In a certain sense, the very premise of science is that one particular reality exists and that we can understand this physical reality through proper scientific investigation and formulate its laws in the language of mathematics.
But this brings back the powerful question from the beginning of this essay: Why is there an asymmetry in physics? Why are these laws of physics the only laws that are real?
Answering this question demands that we examine these laws and see what’s special about them. When we do so, we find that they have two distinct features. First, as Vilenkin said, the vast majority of possible laws are “horrendously large and cumbersome”, in contrast to “the beauty and simplicity of the theories describing our world.” Second, as we explained in Series 1 essay 7, the vast majority of laws lead to a universe with no complexity and structure, in contrast to our actual laws which produce a universe with atoms, molecules, planets, stars, galaxies, and life.
These two features point to a clear and intuitive answer to the question of “Why these laws?” as opposed to any others. Together, they indicate the existence of an intelligent cause that designed these simple and elegant laws for the purpose of producing our amazing complex universe.
However, someone who, like Tegmark, recognizes the fundamental problem of explaining the asymmetry in physics, but automatically rejects the possibility of God, is stuck in a quandary. He rightly doesn’t want to follow the lead of other atheist scientists and say that our laws of physics are arbitrary brute facts of reality with no explanation, but without God, there’s simply no other way of making sense of why only our observed laws of physics are real. Tegmark has no choice but to deny the asymmetry in physics in the first place and say that all laws of physics are actually real just like all mathematical structures have equal mathematical reality. As wild and unlikely as the mathematical multiverse is, there’s really no way, other than God, to truly explain the design of the laws.
By equating physics with mathematics and granting equal physical existence to every possibility that any mathematician can imagine, the mathematical multiverse implicitly undermines the value of all truth that science has discovered, or ever will discover. What truth value do the great scientific discoveries of general relativity and quantum mechanics have if they are just two laws out of the infinite set of mathematical laws that all actually exist?
On the other hand, far from ending scientific investigation, as some atheists wrongfully assert, the theory of an intelligent cause allows science to proceed on a firm logical foundation. The belief in God allows scientists to understand why we observe simple and beautiful laws that produce our complex universe. The belief in God alleviates the need to destroy all scientific truth by positing that all horrendously complex laws that produce meaningless garbage universes are just as real as our beloved laws of nature. The belief in God is the foundation of science, without which science becomes either arbitrary or nonsensical.
The mathematical multiverse’s failure to differentiate between science and mathematics is merely the most extreme and glaring example of the damage that multiverse theories inflict on science. In the next essay, we’ll discuss the scientific method, as it’s been understood by generations of scientists, and show how multiverse theory represents a significant departure from it.
Once we do that, we’ll be in a position to see why multiverse theory is better categorized as speculative science-based philosophy as opposed to the genuine scientific theory its supporters claim it is.
We’ll discuss the major controversy the scientific world is embroiled in as multiverse scientists propose changing the very definition of science to accommodate the multiverse under its esteemed banner. All that and more, so stay tuned!
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