# Origin of “bootstrapping” in mathematical logic

"Bootstrapping" is a term which in general refers to a self-starting process. It is very heavily used in the field of computer science, but it also has uses elsewhere.

For example, in mathematical logic it has been used to refer to the process of taking a set of axioms and then proving useful but tedious lemmas and theorems which will be necessary for future proofs of more interesting theorems. From Samuel Buss' PhD thesis Bounded Arithmetic, dated 1985-1986:

The term "bootstrapping" is a computer term which describes the process of starting the operations of a computer. It used to be common to power up a computer with only a small amount of software loaded, say about 80 bytes, the amount of data which fits on a Hollerith card. This small amount of software would be responsible for reading from tape or cards the entire operating system, thus making the computer fully operational. This process was called "bootstrapping" from the analogy of "lifting oneself by the bootstraps."

Similarly we need to bootstrap $$\mathsf{S_2^1}$$. That is, we shall have to do a lot of work to define some simple functions and predicates in $$\mathsf{S_2^1}$$ (for example, subtraction). Once we have completed the bootstrapping it will be easy to show that $$\mathsf{S_2^1}$$ is actually a fairly strong system which can define a variety of functions and predicates.

I am wondering if either this term had been used before to describe this process, which is ubiquitous in mathematical logic as in for example defining ordinal arithmetic in ZFC so you can prove arithmetical theorems, or if Buss was just borrowing terminology from computer science given the close relationship between the proof theory of bounded arithmetic and computational complexity theory.

For more context which might help, I think that the necessity of bootstrapping axiomatic theories in order to start doing actual work with them and whether or not this is a result of the theories being too generalized is an interesting foundational question. When writing about that question, I would like to be able to say something along the lines of "This process has been referred to as 'bootstrapping', see e.g. [Buss 1986]" so I am interested in the first recorded instances of the term being used.

• "Bootstrapping" was used in other branches of mathematics with similar meaning. MathSciNet finds early uses in mathematical physics as early as 1963. – Gerald Edgar Feb 18 at 1:55
• I don't understand your question. "Bootstrapping" derives from the idiom "Pull yourself up by your bootstraps," suggesting you're so low your boot-tops are higher than you are. Everyone in every branch of anything uses "bootstrap' to suggest starting from nothing. – Carl Witthoft Feb 18 at 14:38
• @CarlWitthoft If everyone in every branch of anything uses it then for each branch someone used it first. You might think it's a trivial or pointless question but I don't understand how you don't understand the question itself. – Not_Here Feb 18 at 18:03

All early mentions on MathSciNet refer to hadron bootstrap models, that "that use very general consistency criteria to determine the form of a quantum theory from some assumptions on the spectrum of particles", e.g. Rockmore Some Equivalent Approaches to the Self-Consistent Bound State, 1963, or computer science/statistics, e.g. Friedman-Rubin On Some Invariant Criteria for Grouping Data, 1967.

"We wish to separate the n objects into g groups in some "optimal'' manner. Some definitions of "optimal'' are proposed and some "bootstrap" (iterative) computer methods are proposed, with some numerical examples. One of the proposed methods is invariant with respect to non-singular linear transformations of the space, but the authors are not dogmatically in favour of any one method."

Bootstrap for elliptic PDE, where one starts from some weak regularity assumptions about solutions and then proves that they are, in fact, much more regular, seemingly out of nothing (actually, because they are elliptic solutions) appears as early as 1972, e.g. in the review of Coffman's On the bifurcation theory of semilinear elliptic eigenvalue problems.

"The author applies a "bootstrap" method in the bifurcation problem for the semilinear elliptic eigenvalue problem... for which the Dirichlet problem is solvable."

Miller compares the idea that there are no atomic facts to hadron bootstrap in a commentary on the Tractatus in 1977 ("The article closes with some perfunctory speculation (put "half seriously, half frivolously'') on the possibility that there are no basic, or atomic, facts. The entertained view is similar to the so-called "nuclear democracy'' or "hadron bootstrap'' view in nuclear physics"), but I do not think this is relevant. It seems that Buss just took it from computer science.

Scare quotes typically indicate novelty, they are absent in Rockmore's case, so hadron model use might have been established by then. As a colloquialism, the expression is much older, although the linkage to Raspe's Baron Munchausen, who "bootstrapped" himself out of the swamp by his own pigtail, appears to be a late addition. According to Wikipedia's etymology:

"The idiom dates at least to 1834, when it appeared in the Workingman's Advocate: "It is conjectured that Mr. Murphee will now be enabled to hand himself over the Cumberland river or a barn yard fence by the straps of his boots."[4] In 1860 it appeared in a comment on philosophy of mind: "The attempt of the mind to analyze itself [is] an effort analogous to one who would lift himself by his own bootstraps.""