Noisy Deductive Reasoning: How Humans Construct Math, and How Math Constructs Universes

Abstract

We present a computational model of mathematical reasoning according to which mathematics is a fundamentally stochastic process. That is, in our model, whether or not a given formula is deemed a theorem in some axiomatic system is not a matter of certainty, but is instead governed by a probability distribution. We then show that this framework gives a compelling account of several aspects of mathematical practice. These include: 1) the way in which mathematicians generate research programs, 2) the applicability of Bayesian models of mathematical heuristics, 3) the role of abductive reasoning in mathematics, 4) the way in which multiple proofs of a proposition can strengthen our degree of belief in that proposition, and 5) the nature of the hypothesis that there are multiple formal systems that are isomorphic to physically possible universes. Thus, by embracing a model of mathematics as not perfectly predictable, we generate a new and fruitful perspective on the epistemology and practice of mathematics.

A   Probabilistic Turing Machines

Perhaps the most famous class of computational machines are Turing machines. One reason for their fame is that it seems one can model any computational machine that is constructable by humans as a Turing machine. A bit more formally, the Church-Turing thesis states that “a function on the natural numbers is computable by a human being following an algorithm, ignoring resource limitations, if and only if it is computable by a Turing machine.”

There are many different definitions of Turing machines (TMs) that are “computationally equivalent” to one another. For us, it will suffice to define a TM as a 7-tuple $(R,\Lambda ,b,v,r^\varnothing ,r^A,\rho )$ where:

• 1. R is a finite set of computational states;

• 2. $\Lambda$ is a finite alphabet containing at least three symbols;

• 3. $b \in \Lambda$ is a special blank symbol;

• 4. $v \in {\mathbb {Z}}$ is a pointer;

• 5. $r^\varnothing \in R$ is the start state;

• 6. $r^A \in R$ is the halt state; and

• 7. $\rho : R \times {\mathbb {Z}}\times \Lambda ^\infty \rightarrow R \times {\mathbb {Z}}\times \Lambda ^\infty$ is the update function. It is required that for all triples (r, v, T), that if we write $(r', v', T') = \rho (r, v, T)$, then $v'$ does not differ by more than 1 from v, and the vector $T'$ is identical to the vectors T for all components with the possible exception of the component with index v⁷;

We sometimes refer to R as the states of the “head” of the TM, and refer to the third argument of $\rho$ as a tape, writing a value of the tape (i.e., of the semi-infinite string of elements of the alphabet) as T.

Any TM $(R,\Sigma ,b,v,r^\varnothing , r^A, \rho )$ starts with $r = r^\varnothing$, the counter set to a specific initial value (e.g, 0), and with T consisting of a finite contiguous set of non-blank symbols, with all other symbols equal to b. The TM operates by iteratively applying $\rho$, until the computational state falls in $r^A$, at which time it stops, i.e., any ID with the head in the halt state is a fixed point of $\rho$.

If running a TM on a given initial state of the tape results in the TM eventually halting, the largest blank-delimited string that contains the position of the pointer when the TM halts is called the TM’s output. The initial state of T (excluding the blanks) is sometimes called the associated input, or program. (However, the reader should be warned that the term “program” has been used by some physicists to mean specifically the shortest input to a TM that results in it computing a given output.) We also say that the TM computes an output from an input. In general, there will be inputs for which the TM never halts. The set of all those inputs to a TM that cause it to eventually halt is called its halting set.

The set of triples that are possible arguments to the update function of a given TM are sometimes called the set of instantaneous descriptions (IDs) of the TM. Note that as an alternative to the definition in (7) above, we could define the update function of any TM as a map over an associated space of IDs.

In one particularly popular variant of this definition of TMs the single tape is replaced by multiple tapes. Typically one of those tapes contains the input, one contains the TM’s output (if and) when the TM halts, and there are one or more intermediate “work tapes” that are in essence used as scratch pads. The advantage of using this more complicated variant of TMs is that it is often easier to prove theorems for such machines than for single-tape TMs. However, there is no difference in their computational power. More precisely, one can transform any single-tape TM into an equivalent multi-tape TM (i.e., one that computes the same partial function), as shown by Arora and Barak [2].

A universal Turing machine (UTM), M, is one that can be used to emulate any other TM. More precisely, in terms of the single-tape variant of TMs, a UTM M has the property that for any other TM $M'$, there is an invertible map f from the set of possible states of the tape of $M'$ into the set of possible states of the tape of M, such that if we:

• 1. apply f to an input string $\sigma '$ of $M'$ to fix an input string $\sigma$ of M;

• 2. run M on $\sigma$ until it halts;

• 3. apply $f^{-1}$ to the resultant output of M;

then we get exactly the output computed by $M'$ if it is run directly on $\sigma '$.

An important theorem of computer science is that there exist universal TMs (UTMs). Intuitively, this just means that there exists programming languages which are “universal”, in that we can use them to implement any desired program in any other language, after appropriate translation of that program from that other language. The physical CT thesis considers UTMs, and we implicitly restrict attention to them as well.

Suppose we have two strings $s^1$ and $s^2$ where $s^1$ is a proper prefix of $s^2$. If we run the TM on $s^1$, it can detect when it gets to the end of its input, by noting that the following symbol on the tape is a blank. Therefore, it can behave differently after having reached the end of $s^1$ from how it behaves when it reaches the end of the first $\ell (s^1)$ bits in $s^2$. As a result, it may be that both of those input strings are in its halting set, but result in different outputs. A prefix (free) TM is one in which this can never happen: there is no string in its halting set that is a proper prefix of another string in its halting set. For technical reasons, it is conventional in the physics literature to focus on prefix TMs, and we do so here.

The coin-flipping distribution of a prefix TM M is the probability distribution over the strings in M’s halting set generated by IID “tossing a coin” to generate those strings, in a Bernoulli process, and then normalizing. So any string $\sigma$ in the halting set has probability $2^{-\;|\;\sigma \;|\;} / \Omega$ under the coin-flipping prior, where $\Omega$ is the normalization constant for the TM in question.

Finally, for our purposes, a Probabilistic Turing Machine (PTM) is a conventional TM as defined by conditions (1)–(7), except that the update function $\rho$ is generalized to be a conditional distribution. The conditional distribution is not arbitrary however. In particular, we typically require that there is zero probability that applying such an update conditional distribution violates condition (7). Depending on how we use a PTM to model NDR machines, we may introduce other requirements as well.