As it stands, there is no single, well-defined philosophical subfielddevoted to the study of non-deductive methods in mathematics. As theterm is being used here, it incorporates a cluster of differentphilosophical positions, approaches, and research programs whosecommon motivation is the view that (i) there are non-deductive aspectsof mathematical methodology and that (ii) the identification andanalysis of these aspects has the potential to be philosophicallyfruitful.

• 1. Introduction
  • 1.1 Discovery versus Justification
  • 1.2 Deduction and Formalization
  • 1.3 Deductivism and Foundations
• 2. Non-deductive Aspects of the Deductive Method
  • 2.1 Aspects of Informality
    • 2.1.1 Semi-Formal Proofs
    • 2.1.2 Gaps in Proofs
    • 2.1.3 Diagrams
  • 2.2 Justifying Deduction
    • 2.2.1 Justification of Rules
    • 2.2.2 The Status of Axioms
  • 2.3 Gödel’s results
• 3. Alternative Non-deductive Methods
  • 3.1 Experimental Mathematics
  • 3.2 Enumerative Induction
  • 3.3 Computer Proofs
  • 3.4 Probabilistic Proofs
• 4. Summary / Conclusions
• Bibliography
• Academic Tools
• Other Internet Resources
• Related Entries

1. Introduction

Philosophical views concerning the ontology of mathematics run thegamut from platonism (mathematics is about a realm of abstractobjects), to fictionalism (mathematics is a fiction whose subjectmatter does not exist), to formalism (mathematical statements aremeaningless strings manipulated according to formal rules), with noconsensus about which is correct. By contrast, it seems fair to saythat there is a philosophically established received view of the basicmethodology of mathematics. Roughly, it is that mathematicians aim toprove mathematical claims of various sorts, and that proof consists ofthe logical derivation of a given claim from axioms. This view has along history; thus Descartes writes in hisRules for the Directionof the Mind (1627–28) that a mathematical proposition mustbe “deduced from true and known principles by the continuous anduninterrupted action of a mind that has a clear vision of each step inthe process” (47). An important implication of this view is thatthere is no room, at least ideally, in mathematics for non-deductivemethods. Frege, for example, states that “it is in the nature ofmathematics always to prefer proof, where proof is possible, to anyconfirmation by induction” (1884, 2). Berry (2016) offers a morerecent defense of proof as promoting key virtues of shared enquirywithin the mathematical community.

In the philosophical literature, perhaps the most famous challenge tothis received view has come from Imre Lakatos, in his influential(posthumously published) 1976 book,Proofs andRefutations:

Euclidean Methodology has developed a certain obligatory style ofpresentation. I shall refer to this as ‘deductiviststyle’. This style starts with a painstakingly stated list ofaxioms, lemmas and/ordefinitions. The axioms anddefinitions frequently look artificial and mystifyingly complicated.One is never told how these complications arose. The list of axiomsand definitions is followed by the carefully wordedtheorems.These are loaded with heavy-going conditions; it seems impossible thatanyone should ever have guessed them. The theorem is followed by theproof.

In deductivist style, all propositions are true and all inferencesvalid. Mathematics is presented as an ever-increasing set of eternal,immutable truths.

Deductivist style hides the struggle, hides the adventure. The wholestory vanishes, the successive tentative formulations of the theoremin the course of the proof-procedure are doomed to oblivion while theend result is exalted into sacred infallibility (Lakatos 1976, 142).

Before proceeding, it will be worthwhile to make a few distinctions inorder to focus the topics of subsequent discussion.

1.1 Discovery versus Justification

The broad claim that there are some non-deductive aspects ofmathematicalactivity seems relatively uncontroversial. Forthis merely amounts to the claim that not everything thatmathematicians do when they do mathematics consists of derivingstatements from other statements. As James Franklin puts it:

Mathematics cannot consist just of conjectures, refutations andproofs. Anyone can generate conjectures, but which ones are worthinvestigating? … Which might be capable of proof by a method inthe mathematician’s repertoire? … Which are unlikely toyield answer until after the next review of tenure? The mathematicianmust answer these questions to allocate his time and effort. (Franklin1987, 2)

One way to narrow the general claim so as to make it more substantiveis make use of the familiar (though not entirely unproblematic)distinction between ‘context of discovery’ and‘context of justification’. On the one hand, thisdistinction may allow the traditional deductivist view to bemaintained in the face of Lakatos’s critique, by arguing thatwhat Lakatos is pointing to concerns the context of discovery inmathematics. Within the context of justification, derivation ofresults from axioms may still be the correct and complete story. Someof the reactions of mathematicians to Lakatos’s views have thischaracter, for example the following remark by Morris Kline in aletter written to Lakatos:

I do believe we need much more literature emphasizing the discoveryside of mathematics. All the emphasis, as you know and as you imply,is on the deductive structure of mathematics and the impression beinggiven to students is that one deduces new conclusions from old ones.^[1]

It is also possible to find passages along similar lines in the workof Pólya, who was a major influence on Lakatos:

Studying the methods ofsolving problems, we perceive anotherface of mathematics. Yes, mathematics has two faces; it is therigorous science of Euclid, but it is also something else. Mathematicspresented in the Euclidean way appears as a systematic, deductivescience, butmathematics in the making appears anexperimental, inductive science. (Pólya 1945, vii)[italics in original]

Conversely, in order to pose a genuine challenge to the familiardeductivist position, the counterclaim needs to be that non-deductivemethods play a role in thejustification of mathematicalresults (Paseau 2015). It will therefore be primarily justificatorycontexts which will be focused on in the remainder of this survey.^[2]

1.2 Deduction and Formalization

This is not the place for a detailed analysis of deduction. Forpresent purposes, this notion will be presumed to be fairlystraightforward, at least in principle. A deduction is any sequence ofstatements each of which is derived from some initial set ofstatements (the premises) or from a prior statement in the sequence.However, one issue that does need to be addressed is the relationshipbetween deduction and formalization (see, e.g., Azzouni 2013).

An argument may be deductive without being formal. Although theparadigm cases of deduction do tend to occur in highly formalizedsystems, this is not necessary. “All even numbers greater than 2are composite; 1058 is greater than 2; 1058 is even; hence 1058 iscomposite” is a perfectly good deduction, despite not beingformalized. Hence, contrary to what is sometimes assumed indiscussions of these issues, it is not true that all informal aspectsof mathematical practice arethereby non-deductive. Acorresponding point applies to the notion of rigor. In a recent book(2024), Tanswell argues for a view which he calls “rigorpluralism,” according to which multiple different models ofinformal proof serve to account for different features of the conceptof rigor.

On the other hand, the development of formal logic has been closelybound up with providing a clear language for presenting (andevaluating) deductive mathematical reasoning. Indeed, as John Burgessargues in his (1992), modern classical logic largely developed as abasis for mathematical reasoning, especially proof. The increase inrigor within mathematics during the 19^th Century isproperly viewed as a cause, not an effect, of the logical revolutionset off by Frege’s work. Logic, in Burgess’s view, isdescriptive: its goal is to construct mathematical models ofreasoning. Classical logic constitutes an idealized description ofclassical mathematical proof.

It may also be important to distinguishinformal elements ofa given mathematical proof fromunformalizable elements (ifthere are any such things).^[3] In Section 2.1.3, this issue will be taken up in connection with theuse of diagrams in mathematical reasoning.

1.3 Deductivism and Foundations

In addition to the development of formal logic, another aspect ofdeductivism is its emphasis on ‘foundations’. The reasonfor this is that the passage from axioms to theorem isstraightforward, in principle, since it is a matter of logicalderivation. Indeed there is nothing distinctively mathematicalinvolved in this transition. Hence attention is shifted to thestarting point of the deductive process, namely the axioms. And ifthese axioms are themselves theorems of some more basic theory, thenthis pursuit of a secure starting point can be pursued down through ahierarchy of ever more foundational mathematical theories.

It is undeniable that issues in the foundations of mathematics havebeen the central preoccupation of philosophers of mathematics throughmost of the 20^th Century. This is not, of course, becausefoundational areas such as set theory are the only areas ofmathematics where philosophers think that deduction takes place, butrather because—as pointed out above—focusing on deductionputs particular emphasis on the starting points of proofs. Even thosesympathetic with this focus on foundational issues are likely toacknowledge that many areas of mathematical practice are therebyignored. The question is what—if anything—of philosophicalinterest is lost in the process.

2. Non-deductive Aspects of the Deductive Method

2.1 Aspects of Informality

2.1.1 Semi-Formal Proofs

As mentioned in 1.2 above, one feature of the deductivist style isthat paradigmatic mathematical proofs are expressed entirely in someappropriate formal language (for example, first-order predicate logicwith identity). This allows the validity of a given proof to beeasily, indeed mechanically, ascertained. But of course few, if any,of the proofs circulated and published by mathematicians have thisform. What qualifies as a proof for working mathematicians ranges fromthe completely informal to the detailed and precise, with every (oralmost every) gap filled in. However, even detailed and precise proofsare rarely expressed purely in the language of logic; rather, they area mixture of ordinary language, mathematical, and logical symbols andterminology.

Sometimes philosophers writing in the deductivist tradition make itsound as if this is a fairly trivial point; it is just a matter ofmathematicians having a ‘translation scheme’ to hand, butnot writing out the proof in pure logic to make it more accessible andeasier to read. In point of fact, it is often far from obvious how totranslate a given proof into formal logic. Furthermore, it is notclear that the notion of ‘translating’ an informal proofinto a formal language is necessarily the right way of looking at thesituation. Stewart Shapiro presents essentially this view at thebeginning of his 1991 book,Foundations WithoutFoundationalism, writing that:

The languages of full logics are, at least in part, mathematicalmodels of fragments of ordinary natural languages, like English, orperhaps ordinary languages augmented with expressions used inmathematics. The latter may be called ‘natural languages ofmathematics’. For emphasis, or to avoid confusion, the languageof a full logic is sometimes called a ‘formal language’.

As a mathematical model, there is always a gap between the language ofa logic and its natural language counterpart. The fit between modeland modelled can be good or bad, useful or misleading, for whateverpurpose at hand. (Shapiro 1991, 3)

An alternative picture is that the formal and informal languages offerdifferent ways of expressing mathematical theorems and proofs. Theformal language is not used to ‘translate’, and hence doesnot need to be measured against what is expressed in an informalproof. Rather it offers its own, arguably superior, resources forexpressing the content of mathematical statements in a precise andrigorous setting that has been specifically designed for this purpose.Whichever picture is adopted of the relation between formal andinformal presentations of mathematics, two points remain. First,deductive mathematical arguments—arguments that are produced,transmitted, and built upon by mathematicians—can be eitherformal or informal. Second, the evaluation of such arguments as beingdeductively valid or invalid is easier to carry out definitively inthe context of a formal system of some sort.

It is also worth noting that Lakatos argues for a third category ofproof, in addition to formal and informal, that he calls“quasi-formal”. Lakatos writes that:

to suggest that an informal proof is just an incomplete formal proofseems to me to make the same mistake as early educationalists didwhen, assuming that a child was merely a miniature grown-up, theyneglected the direct study of child-behaviour in favour of theorizingbased on simple analogies with adult behaviour. (Lakatos 1980, 63)

2.1.2 Gaps in Proofs

The talk above of “every gap being filled in” in thetransition to an ideal proof glosses over the fact that the notion ofa “gap” in a proof is itself in need of furtherclarification. For one thing, the most straightforward way of defininga proof gap—as given below—is only applicable to fullyformal systems.

Agap is any point in a proof where the line written does notfollow from some subset of the previous lines (together with axioms)by the application of a formally valid—and explicitlystated—rule of inference for the system.

The reason for the condition that any rule be an explicitly statedrule of inference for the system is because we want to make room forgappy yet valid proofs. For example, “2 + 2 = 4, hence there areinfinitely many primes” is a valid argument, but clearly thereis a large gap between its premise and its conclusion. On the otherhand, despite the above definition only working for formal proofs,gaplessness and formality do not always go together. Thus atraditional syllogism such as, “All men are mortal; Socrates isa man; hence Socrates is mortal” is an example of a gaplessinformal proof. One way to extend the notion of gappiness (andgaplessness) to informal proofs is via the notion of abasicmathematical inference, in other words an inference that is“accepted by the mathematical community as usable in proofwithout any further need of argument” (Fallis 2003, 49).

However we end up characterizing gaps, it is undeniably the case thatmost actual proofs as presented by mathematicians have gaps. DonFallis proposes a taxonomy of kinds of proof gaps in his (2003):

1. Inferential Gaps
   “A mathematician has left an inferential gap whenever theparticular sequence of propositions that the mathematician has in mind(as being a proof) is not a proof” (Fallis 2003, 53).
2. Enthymematic Gaps
   “A mathematician has left an enthymematic gap whenever he doesnot explicitly state the particular sequence of propositions that hehas in mind” (Fallis 2003, 54).^[4]
3. Untraversed Gaps
   “A mathematician has left an untraversed gap whenever he has nottried to verify directly that each proposition in the sequence ofpropositions that he has in mind (as being a proof) follows fromprevious propositions in the sequence by a basic mathematicalinference” (Fallis 2003, 56–7).

In addition to this taxonomical work, Fallis also argues for thephilosophical thesis that gaps in proofs are not necessarily a badthing. Building on (iii) above, he introduces the notion of auniversally untraversed gap, in other words a gap that hasnot been bridged by any member of the mathematical community. Fallisclaims that such gaps are not unusual and that at least some of thetime proofs containing them are accepted by mathematicians in ajustificatory context. This view is borne out in more recent work byAndersen (2020).

One currently active area of work that has led to the uncovering ofhitherto unrecognized gaps of various sorts is automated proofchecking. Specially designed computer programs are used to check thevalidity of proofs that have been rendered in an appropriate formallanguage. The main focus thus far has not been on discovering newresults but on checking the status of proofs of already establishedresults. George Gonthier has used this approach to verify a proof ofthe four color theorem (Gonthier 2008) and a proof of the odd ordertheorem in group theory (Gonthier et al. 2013), and Thomas Hales hasverified a proof of the Jordan curve theorem (Hales 2007). In eachcase, a number of gaps were found and then traversed. Formalverification of this sort can also reveal other information hidden inthe content of ordinary mathematical arguments. Georg Kreisel hasdescribed this general process as “unwinding proofs”,while Ulrich Kohlenbach has more recently coined the term “proofmining.” In connection with the methods described above, Avigadwrites that

… proof-theoretic methods and insights can be put to use… in the field of automated reasoning and formal verification.Since the early twentieth century, it has been understood thatordinary mathematical arguments can be represented in formal axiomatictheories, at least in principle. The complexity involved in even themost basic mathematical arguments, however, rendered mostformalization infeasible in practice. The advent of computationalproof assistants has begin to change this, making it possible toformalize increasingly complex mathematical proofs. … [T]hemethods can also be used for the more traditional task of verifyingordinary mathematical proofs, and are especially pertinent to caseswhere proofs rely on computation that is too extensive to check byhand. (Avigad 2007, 7)

Delariviere and Van Kerkhove (2017) point out, however, that whilecomputer methods may play an increasingly important rule in proofverification, it is much less clear that such methods can play acorrespondingly central role in advancing mathematicalunderstanding.

2.1.3 Diagrams

Another aspect of informal proof that has been the subject of renewedattention in the recent philosophical literature is the role ofdiagrams (Giaquinto 2007; Shin & Lemon 2008). What is not indispute is that proofs—especially in geometry but also in otherareas ranging from analysis to group theory—are oftenaccompanied by diagrams. One issue concerns whether such diagrams playan indispensable role in the chain of reasoning leading from thepremises of a given proof to its conclusion.Prima facie,there would seem to be three possible situations:

1. The diagrams play no substantive role in the proof and servemerely as ‘illustrations’ of aspects of the subject-matterwith which it deals.
2. In practical terms it is difficult (or even impossible) to graspthe proof without making use of the diagrams, but thisindispensability is psychological rather than logical.
3. The diagrams play an essential role in the logical structure ofthe proof.

The initial wave of philosophical work done on diagrammatic reasoningfocused on Euclid’sElements, partly because of thecentrality and historical importance of this work, and partly becauseit is so often held up as a canonical example of the deductive method(see, e.g., Mumma 2010). If some or all of the diagrams in theElements fall under option (iii) above, then deleting all thediagrams will render many of the proofs invalid. This raises thefurther question of whether a distinctively diagrammatic form ofreasoning can be identified and analyzed, and—ifso—whether it can be captured in a purely deductive system. Onedifficulty for any proposed rigorization is the ‘generalizationproblem’: how can a proof that is linked to a specific diagrambe generalized to other cases? This is intertwined with the issue ofdistinguishing, in formal terms, between the essential andcoincidental features of a given diagram.

More recent work on the role of diagrams in proofs has included adefense of the position that diagrammatic proofs can sometimes befully rigorous (Azzouni, 2013), investigation of diagram-basedreasoning in areas of mathematical practice other than geometry (deToffoli and Giardino, 2014; de Toffoli, 2017), and exploration of howdiagrams may be more actively involved in the reasoning processthrough the concept of “epistemic action” (Novaes2025).

2.2 Justifying Deduction

Even if we restrict attention to the context of justification, adeductive proof yields categorical knowledge only if it proceeds froma secure starting point and if the rules of inference aretruth-preserving. Can our confidence that these two conditions obtainalso be grounded purely deductively? These conditions will beconsidered in turn.

2.2.1 Justification of Rules

In one sense, it seems quite straightforward to give a deductivejustification for some favored set of rules of inference. It can beshown, for example, that if the premises of an application of ModusPonens are true then the conclusion must also be true. The problem, atleast potentially, is that such justifications typically make use ofthe very rule which they seek to justify. In the above case: if MP isapplied to true premises then the conclusion is true; MP is applied totrue premises; hence the conclusion is true. Haack (1976) and othershave debated whether the circularity here is vicious or not. Onecrucial consideration is whether analogous‘justifications’ can be given for invalid rules, forexample Prior’s introduction and elimination rules for‘tonk,’ that also have this feature of using a rule tojustify itself.^[5] (A closely related issue can be traced back to Lewis Carroll and hisclassic (1895) paper.)

2.2.2 The Status of Axioms

Let us assume, then, that an idealized deductive proof provides onekind of security: the transparency of each step ensures the validityof the argument as a whole, and hence guarantees thatif thepremises are all truethen the conclusion must be true. Butwhat of the axioms that are brought in at the beginning of the proofprocess? The traditional answer to this question is to claim that thetruth of the axioms is secure because the axioms are“self-evident”. This certainly seems to have been thegenerally accepted view of the axioms of Euclidean geometry, forexample. However, this attitude is much less prevalent in contemporarymathematics, for various reasons. Firstly, the discovery ofnon-Euclidean geometry in the early 19^th Century showedthat apparent self-evidence, at least in the case of the ParallelPostulate, is no guarantee of necessary truth. Secondly, theincreasing range and complexity of mathematical theories—andtheir axiomatizations—made it much less plausible to claim thateach individual axiom was transparently true. Thirdly, manymathematical subfields have become abstracted to a considerable degreefrom any concrete models, and this has gone hand-in-hand with thetendency for at least some mathematicians to adopt a formalistattitude to the theories they develop. Rather than expressingfundamental truths, on this view axioms serve simply to provide thestarting position for a formal game.

The slide towards this sort of formalist attitude to axioms can alsobe traced through Frege’s logicism. The logicist program soughtto show that mathematics is reducible to logic, in other words thatmathematical proofs can be shown to consist of logical deductions fromlogically true premises. For Frege, these logically true premises aredefinitions of the terms which occur in them. But this againraises the issue of what distinguishes acceptable from unacceptabledefinitions. The worry here is not just whether our axioms are truebut whether they are even consistent (a pitfall which famously befellFrege’s own system). And this is a problem once self-evidence isabandoned as the ‘gold standard’ for axioms, whether wemove from here to a formalist view or a logicist view. In both cases,some other bounds on the acceptability of candidate axioms must beprovided.

Is there a middle ground, then, between the high standard ofself-evidence on the one hand and the ‘anything goes’attitude on the other? One idea, a version of which can be traced backto Bertrand Russell, is to invoke a version of inference to the bestexplanation. Russell’s view, plausibly enough, is that thepropositions of elementary arithmetic—“2 + 2 = 4”,“7 is prime”, etc.—are much more self-evident thanthe axioms of whatever logical or set-theoretic system one might comeup with to ground them. So rather than viewing axioms as maximallyself-evident we ought instead to think of them as being chosen on thebasis of their (collective) capacity to systematize, derive, andexplain the basic arithmetical facts. In other words, the direction oflogical implication remains from axioms to arithmetical facts, but thedirection of justification may go the other way, at least in the caseof very simple, obvious arithmetical facts. Deriving “2 + 2 =4” from our set-theoretic axioms does not increase ourconfidence in the truth of “2 + 2 = 4”, but the fact thatwe can derive this antecedently known fact (and not derive otherpropositions which we know to be false) does increase our confidencein the truth of the axioms.

The direction of justification here mirrors the direction ofjustification in inference to the best explanation. Once we have ameasure of confidence in a particular choice of axioms then thedirection of justification can also flow in the more conventionaldirection, in step with the deductive inferences of a proof. This willhappen when the theorem proved was not one whose truth wasantecedently obvious. Easwaran (2005), Mancosu (2008), and Schlimm(2013) have developed this basic account of axiom choice in differentways. For example, Mancosu argues that an analogous process mayunderlie the development of new mathematical theories that extend thedomain of application or the ontology of previous theories. Makingfurther progress on analyzing this process will depend on giving asatisfactory account of mathematical explanation, and this has becomean area of considerable interest in the recent literature onphilosophy of mathematics.

Another approach, pursued by Maddy (1988, 1997, 2001, 2011) is to lookin more detail at the actual practice of mathematicians and thereasons they give for accepting or rejecting different candidateaxioms. Maddy’s main focus is on axioms for set theory, and sheargues that there are various theoretical virtues, with no direct linkto ‘self-evidence’, which axioms may possess. What thesevirtues are, and how they are weighted relative to one another, maywell vary across different areas of mathematics. Two core virtueswhich Maddy identifies for set-theoretic axioms are UNIFY (i.e. thatthey provide a single foundational theory for deciding set-theoreticquestions) and MAXIMIZE (i.e. that they not arbitrarily restrict therange of isomorphism types). The issue of axiom choice in set theoryhas also been taken up in recent work by Lingamneni (2020) and byFontanella (2019).

2.3 Gödel’s results

Undoubtedly the most notorious of the limitations on the deductivemethod in mathematics are those which stem from Gödel’sincompleteness results. Although these results apply only tomathematical theories strong enough to embed arithmetic, thecentrality of the natural numbers (and their extensions into therationals, reals, complexes, etc.) as a focus of mathematical activitymeans that the implications are widespread.

Nor should the precise implications of Gödel’s work beoverstated. The order of the quantifiers is important. What Gödelshowed is that, for any consistent, recursively axiomatized formalsystem, F, strong enough for arithmetic, there are truths expressiblein purely arithmetical language which are not provable in F. He didnot show that there are arithmetical truths which are unprovable inany formal system. Nonetheless, Gödel’s resultsdid hammer some significant nails into the coffin of one version ofthe deductive ideal of mathematics. There cannot be a single,recursively axiomatizable formal system for all of mathematics whichis (a) consistent, (b) purely deductive, and (c) complete. One line ofresponse to this predicament is to explore options for non-deductivemethods of justification in mathematics.

3. Alternative Non-deductive Methods

3.1 Experimental Mathematics

The role of non-deductive methods in empirical science is readilyapparent and relatively uncontroversial (pace Karl Popper).Indeed the canonical pattern of justification in science isaposteriori and inductive. What makes empirical science empiricalis the crucial role played by observation, and—inparticular—by experiment. A natural starting point, therefore,in a survey of non-deductive methods in mathematics, is to look at therise of a genre known as “experimental mathematics.” Thepast 15 years or so have seen the appearance of journals (e.g.,The Journal of Experimental Mathematics), institutes (e.g.,the Institute for Experimental Mathematics at the University ofEssen), colloquia (e.g., the Experimental Mathematics Colloquium atRutgers University), and books (e.g., Borwein and Bailey 2003 and2004) devoted to this theme. These latter authors also argue, inBorwein and Bailey (2015), for the significance of experimentalmathematics within mathematical practice more generally, whileSorensen (2016) provides a broader historical and sociologicalanalysis of experimental mathematics.

Against the background of the traditional dichotomy betweenmathematical and empirical routes to knowledge, the very term“experimental mathematics” seems at best oxymoronic and atworst downright paradoxical. One natural suggestion is thatexperimental mathematics involves performingmathematicalexperiments, where the term “experiment” here isconstrued as literally as possible. This is the approach adopted byvan Bendegem (1998). According to van Bendegem, an experiment involves“the manipulation of objects, … setting up processes inthe ‘real’ world and … observing possible outcomesof these processes” (Van Bendegem 1998, 172). His suggestion isthat the natural way to get an initial grip on what a mathematicalexperiment might be is to consider how an experiment in thisparadigmatic sense might have mathematical ramifications.

One example that van Bendegem cites dates back to work done by the19^th-century Belgian physicist Plateau on minimal surfacearea problems. By building various geometrical shapes out of wire anddipping these wire frames into a soap solution, Plateau was able toanswer specific questions about the minimum surface bounding variousparticular shapes, and—eventually—to formulate somegeneral principles governing the configurations of such surfaces.^[6] One way of understanding what is going on in this example is that aphysical experiment—the dipping of a wire frame into a soapsolution—is producing results that are directly relevant to acertain class of mathematical problem. The main drawback of this wayof characterizing experimental mathematics is that it is toorestrictive. Examples of the sort van Bendegem cites are extremelyrare, hence the impact of mathematical experiments of this sort onactual mathematical practice can only be very limited at best.Moreover, it cannot be only this, literal sense of experiment thatmathematicians have in mind when they talk about—anddo—experimental mathematics.

So much for the most literal reading of “mathematicalexperiment.” A potentially more fruitful approach is to think inanalogical or functional terms. In other words, perhaps“experimental mathematics” is being used to labelactivities which function within mathematics in a way analogous to therole of experiment in empirical science. Thus mathematical experimentsmay share some features with literal experiments, but not otherfeatures (Baker 2008; McEvoy 2008, 2013; Sorensen 2010; van Kerkhove2008). Insights may also be gained by looking at the historicaldevelopment of the notion of a mathematical experiment (Sorensen2024), and at potential links between mathematical experiments andthought experiments (Giardino, 2022). Before proceeding with thisbroad line of analysis, it may be helpful to look briefly at a casestudy.

A nice example of contemporary work in experimental mathematicsappears in a book by Borwein and Bailey (1995b, Ch. 4). A real numberis said to benormal in base n if every sequence of digitsfor base n (of any given length) occurs equally often in its base-nexpansion. A number isabsolutely normal if it is normal inevery base. Consider the following hypothesis:

Conjecture: Every non-rational algebraic number is absolutely normal.

Borwein and Bailey used a computer to compute to 10,000 decimal digitsthe square roots and cube roots of the positive integers smaller than1,000, and then they subjected these data to certain statisticaltests.

There are a couple of striking features of this example that may pointto a more general characterization of experimental mathematics.Firstly, the route from evidence to hypothesis is via enumerativeinduction. Secondly, it involves the use of computers. In whatfollows, these two features will be examined in turn, both to helpilluminate the notion of experimental mathematics, and asnon-deductive methods in their own right.

3.2 Enumerative Induction

In a letter to Euler written in 1742, Christian Goldbach conjecturedthat all even numbers greater than 2 are expressible as the sum of two primes.^[7] Over the following two and a half centuries, mathematicians have beenunable to prove Goldbach’s Conjecture. However, it has beenverified for many billions of examples, and there appears to be aconsensus among mathematicians that the conjecture is most likelytrue. Below is a partial list showing the order of magnitude up towhich all even numbers have been checked and shown to conform toGC.

Despite this vast accumulation of individual positive instances of GC,aided since the early 1960s by the introduction—and subsequentrapid increases in speed—of the digital computer, no proof of GChas yet been found. Not only this, but few number theorists areoptimistic that there is any proof in the offing. Fields medalist AlanBaker stated in a 2000 interview: “It is unlikely that we willget any further [in proving GC] without a big breakthrough.Unfortunately there is no such big idea on the horizon.” Also in2000, publishers Faber and Faber offered a $1,000,000 prize to anyonewho proved GC between March 20 2000 and March 20 2002, confident thattheir money was relatively safe.

What makes this situation especially interesting is thatmathematicians have long been confident in the truth of GC. Hardy& Littlewood asserted, back in 1922, that “there is noreasonable doubt that the theorem is correct,” and Echeverria,in a recent survey article, writes that “the certainty ofmathematicians about the truth of GC is complete” (Echeverria1996, 42). Moreover, this confidence in the truth of GC is typicallylinked explicitly to the inductive evidence: for instance, G.H. Hardydescribed the numerical evidence supporting the truth of GC as“overwhelming.” Thus it seems reasonable to conclude thatthe grounds for mathematicians’ belief in GC is the enumerativeinductive evidence.

One distinctive feature of the mathematical case which may make adifference to the justificatory power of enumerative induction is theimportance of order. The instances falling under a given mathematicalhypothesis (at least in number theory) are intrinsically ordered, andfurthermore position in this order can make a crucial difference tothe mathematical properties involved. As Frege writes, with regard tomathematics:

[T]he ground [is] unfavorable for induction; for here there is none ofthat uniformity which in other fields can give the method a highdegree of reliability. (Frege,Foundations of Arithmetic)

Frege then goes on to quote Leibniz, who argues that difference inmagnitude leads to all sorts of other relevant differences between thenumbers:

An even number can be divided into two equal parts, an odd numbercannot; three and six are triangular numbers, four and nine aresquares, eight is a cube, and so on. (Frege,Foundations ofArithmetic)

Frege also explicitly compares the mathematical and non-mathematicalcontexts for induction:

In ordinary inductions we often make good use of the proposition thatevery position in space and every moment in time is as good in itselfas every other. … Position in the number series is not a matterof indifference like position in space. (Frege,Foundations ofArithmetic)

As Frege’s remarks suggest, one way to underpin an argumentagainst the use of enumerative induction in mathematics is via somesort ofnon-uniformity principle: in the absence of proof, weshould not expect numbers (in general) to share any interestingproperties. Hence establishing that a property holds for someparticular number gives no reason to think that a second, arbitrarilychosen number will also have that property.^[8] Rather than the Uniformity Principle which Hume suggests is the onlyway to ground induction, we have almost precisely the oppositeprinciple! It would seem to follow from this principle thatenumerative induction is unjustified, since we should not expect(finite) samples from the totality of natural numbers to be indicativeof universal properties.

A potentially even more serious problem, in the case of GC and in allother cases of induction in mathematics, is that the sample we arelooking at isbiased. Note first thatall knowninstances of GC (and indeed all instances it is possible to know)are—in an important sense—small.

In a very real sense, there are no large numbers: Any explicit integercan be said to be “small”. Indeed, no matter how manydigits or towers of exponents you write down, there are only finitelymany natural numbers smaller than your candidate, and infinitely manythat are larger (Crandall and Pomerance 2001, 2).

Of course, it would be wrong to simply complain that all instances ofGC arefinite. After all, every number is finite, so if GCholds for all finite numbers than GC holdssimpliciter.^[9] But we can isolate a more extreme sense of smallness, which might betermedminuteness (Baker 2007).

Definition: a positive integer,n, isminute just incasen is within the range of numbers we can write down usingordinary decimal notation, including (non-iterated) exponentiation.

Verified instances of GC to date are not just small, they are minute.And minuteness, though admittedly rather vaguely defined, is known tomake a difference. Consider, for example, the logarithmic estimate ofprime density (i.e. the proportion of numbers less than a givenn that are prime) which is known to become an underestimatefor large enoughn. Letn^* be the firstnumber for which the logarithmic estimate is too small. If the RiemannHypothesis is true, then it can be proven that an upper bound forn^* (the first Skewes number) is 8 ×10³⁷⁰. Though an impressively large number, it isnonetheless minute according to the above definition. However if theRiemann Hypothesis is false than our best known upper bound forn^* (the second Skewes number) is 10↑10↑10↑10↑3.^[10] The necessity of inventing an ‘arrow’ notation here torepresent this number tells us that it is not minute. The second partof this result, therefore, although admittedly conditional on a resultthat is considered unlikely (viz. the falsity of RH), implies thatthere is a property which holds of all minute numbers but does nothold for all numbers. Minuteness can make a difference.

What about the seeming confidence that number theorists have in thetruth of GC? Echeverria (1996) discusses the important role played byCantor’s publication, in 1894, of a table of values of theGoldbach partition function, G(n), forn = 2 to1,000 (Echeverria 1996,29–30). The partition function measuresthe number of distinct ways in which a given (even) number can beexpressed as the sum of two primes. Thus G(4) = 1, G(6) = 1, G(8) = 1,G(10) = 2, etc. This shift of focus onto the partition functioncoincided with a dramatic increase in mathematicians’ confidencein GC. What became apparent from Cantor’s work is thatG(n) tends to increase asn increases. Note thatwhat GC amounts to in this context is that G(n) never takesthe value 0 (for any evenn greater than 2). The overwhelmingimpression made by data on the partition function is that itis highly unlikely for GC to fail for some largen. Forexample, for numbers on the order of 100,000, there is always at least500 distinct ways to express each even number as the sum of twoprimes!

However, as it stands these results are purely heuristic. The thirtyyears following Cantor’s publication of his table of values(described by Echeverria as the “2^nd period” ofresearch into GC) saw numerous attempts to find an analytic expressionfor G(n). If this could be done then it would presumably becomparatively straightforward to prove that this analytic functionnever takes the value 0 (Echeverria 1996, 31). By around 1921,pessimism about the chances of finding such an expression led to achange of emphasis, and mathematicians started directing theirattention to trying to find lower bounds for G(n). This toohas proved unsuccessful, at least to date.

Thus consideration of the partition function has not brought a proofof GC any closer. However, it does allow us to give an interestingtwist to the argument given above. The graph suggests that the hardesttest cases for GC are likely to occur among the smallest numbers;hence the inductive sample for GCis biased, but it is biasedagainst the chances of GC. Baker (2024) argues that this kindof ‘positive’ size-bias applies more generally, and doesnot require specific arguments of the sort given above for theGoldbach Conjecture. The reason is that small numbers are more likelyto be boundary cases of significant mathematical properties, and theyare also more likely to possess combinations of interestingmathematical properties. Both of these facts make it more likely thatany counterexample to a given conjecture will occur in a relativelysmall initial segment of the natural numbers. Thus the claim about(unavoidable) bias can be embraced without undermining the rationaljustification of enumerative induction for arithmetical conjectures.(See also Paseau 2021.)

3.3 Computer Proofs

A striking feature of contemporary work in experimental mathematics isthat it is doneusing computers. Is this reliance oncomplex pieces of electronics what makes the field‘experimental’? If one looks at what gets published incontemporary journals, books, and conferences devoted to experimentalmathematics, the impression is that all the items are closely bound upwith computers. For example, there does not appear to be a singlepaper published in more than a decade’s worth of issues ofExperimental Mathematics that does not involve the use ofcomputers. What about the kinds of examples which mathematicians tendto offer as paradigms of experimental mathematics? Here the data isless clear. On the one hand, an informal survey suggests that themajority of such exemplars do involve the explicit use of computers.On the other hand, it is not uncommon for mathematicians also to citeone or more historical examples, from well before the computer age, toillustrate the purported pedigree of the subdiscipline.

The biggest practice-based challenge to equating experimentalmathematics with computer-based mathematics comes from whatself-styled experimental mathematicians say about their nascentdiscipline. For when mathematicians self-consciously reflect on thenotion of experimental mathematics, they tend to reject the claim thatcomputer use is a necessary feature. For example, the editors of thejournalExperimental Mathematics—in their“statement of philosophy” concerning the scope and natureof the journal—make the following remarks:

The word “experimental” is conceived broadly: manymathematical experiments these days are carried out on computers, butothers are still the result of pencil-and-paper work, and there areother experimental techniques, like building physical models.(“Aims and Scope”,ExperimentalMathematics—see Other Internet Resources)

And here is another passage with a similar flavor from mathematicianDoron Zeilberger:

[T]raditional experimental mathematics … has been pursued byall the great, and less-great, mathematicians through the centuries,using pencil-and-paper. (Gallian and Pearson 2007, 14)

It seems fair to say that tying experimental mathematics to computeruse fits well with what contemporary experimental mathematicians dobut not so well with what they say.^[11]

In the current context, the central question is not whethercomputer-based mathematics is ‘experimental’ but whetherit is—at least sometimes—non-deductive. In onesense, of course, all of the individual calculations performed by acomputer are deductive, or at least they are isomorphic to theoperations of a purely deductive formal system. When a computerverifies an instance of GC, this verification is completely deductive.We may then separate out two distinct questions. Firstly, are thesecomputations playing a non-deductive role in some larger mathematicalargument? And, secondly, are the beliefs we form directly from theresults of computer computations deductively grounded beliefs? Thefirst of these questions does not turn on anything specific tocomputers, and hence collapses back to the issue discussed in Section3.2 above on enumerative induction. The second question will beexamined below.

Philosophical discussion of the status of computer proofs was promptedin large part by Appel and Haken’s computer-based proof of theFour Color Theorem in 1976. In his (1979), Tymoczkoargues—controversially—that mathematical knowledge basedon computer proofs is essentially empirical in character. This isbecause such proofs are nota priori, not certain, notsurveyable, and not checkable by human mathematicians. In all theserespects, according to Tymoczko, computer proofs are unliketraditional ‘pencil-and-paper’ proofs. Concerningsurveyability, Tymoczko writes:

A proof is a construction that can be looked over, reviewed, verifiedby a rational agent. We often say that a proof must be perspicuous, orcapable of being checked by hand. It is an exhibition, a derivation ofthe conclusion, and it needs nothing outside itself to be convincing.The Mathematiciansurveys the proof in its entirety andthereby comes toknow the conclusion. (Tymoczko 1979, 59)

Assume for sake of argument that the computer proof in question isdeductively correct but is also unsurveyable in the above sense. Doesour decision to rely on the output of the computer here constitute anon-deductive method? One way of viewing this kind of exampleis as driving a wedge between a deductive method and our non-deductiveaccess to the results of that method. Compare, for instance,being told of a particular mathematical result by an expertmathematician (with a good track record). Is this a‘non-deductive method’?^[12]

3.4 Probabilistic Proofs

There is a small, but growing, subset of mathematical methods whichare essentially probabilistic in nature. In the context ofjustification, these methods do not deductively imply their conclusionbut rather establish that there is some (often precisely specifiable)high probability that the conclusion is true. Philosophical discussionof these methods began with Fallis (1997, 2002), while Berry (2019) isa useful recent contribution to the debate.

One type of probabilistic method links back to the earlier discussionof experimental mathematics in that it involves performing experimentsin a quite literal sense. The idea is to harness the processing powerof DNA to effectively create a massively parallel computer for solvingcertain otherwise intractable combinatorial problems. The most famousof these is the ‘Traveling Salesman’ problem, whichinvolves determining whether there is some possible route through thenodes of a graph connected by unidirectional arrows that visits eachnode exactly once. Adleman (1994) shows how the problem can be codedusing strands of DNA which can then be spliced and recombined usingdifferent chemical reactions. The appearance of certain longer DNAstrands at the end of the process corresponds to the finding of asolution path through the graph. Probabilistic considerations come inmost clearly in the case where no longer DNA strands are found. Thisindicates that there is no path through the graph, but even if theexperiment is carried out correctly the support here falls short offull certainty. For there is a small chance that there is a solutionbut that it fails to be coded by any DNA strand at the start of theexperiment.

There are also probabilistic methods in mathematics which are notexperimental in the above sense. For example, there are properties ofcomposite (i.e. non-prime) numbers which can be shown to hold inrelation to about half of the numbers less than a given compositenumber. If various numbers smaller thanN are selected atrandom and none of them bear this relation toN, then itfollows thatN is almost certainly prime. The level ofprobability here can be precisely calculated, and can be made as highas needed by picking more ‘witness’ numbers to test.

Note that these sorts of probabilistic methods contain plenty ofpurely deductive reasoning. Indeed, in the second example, the factthat the probability ofN being prime is .99 is establishedpurely deductively. Nonetheless, there is general consensus in themathematical community that such methods are not acceptablesubstitutes for deductive proof of the conclusion. Fallis (1997, 2002)argues that this rejection is not reasonable because any property ofprobabilistic methods that can be pointed to as being problematic isshared by some proofs that the mathematical community does accept.Fallis’s focus is on establishing truth as the key epistemicgoal of mathematics. However it seems plausible that one major reasonfor mathematicians’ dissatisfaction with probabilistic methodsis that they do not explainwhy their conclusions are true.In addition, Easwaran argues, against Fallis, that there is aproperty, which he terms ‘transferability’, thatprobabilistic proofs lack and acceptable proofs have (Easwaran 2009;Jackson 2009). Fallis (2011) is a reply to some of theseobjections.

On the other hand, there may be cases where the bare truth or falsityof a claim is important even in the absence of accompanyingexplanation. For example, one could imagine a situation in which animportant and interesting conjecture—say the RiemannHypothesis—is being considered, and a probabilistic method isused to show that some number is very likely a counterexample to it.It is interesting to speculate what the reaction of the mathematicalcommunity might be to this situation. Would work on trying to prove RHcease? Would it continue until a rigorous deductive proof of thecounterexample is constructed?

4. Summary / Conclusions

It is not clear why one should expect the various non-deductivemethods used in mathematics to share any substantive features otherthan their non-deductiveness. Philosophers looking at the role ofnon-deductive reasoning in the context of discovery have often talkedas if there is some unity to be found (for example, the subtitle toLakatos’sProofs and Refutations is “the Logic ofMathematical Discovery.” More likely is that the array ofnon-deductive methods is diverse and heterogeneous. (Compare StanislawUlam’s remark that “the study of non-linear physics islike the study of non-elephant biology.”)

Work by contemporary philosophers of mathematics is continuing to pushthe study of non-deductive mathematical methods in new directions. Onearea of interest is in ‘mathematical natural kinds’ andwhether such a notion can be used to ground the use of analogy inmathematical reasoning (Corfield 2004 [Other Internet Resources]).Another area being investigated is the putative role of heuristicprinciples in mathematics. (Much of this work traces back toPólya 1945.) A third area of recent work concerns the role ofBayesianism as a potential framework for thinking about non-deductivereasoning in mathematics. Significant contributions to this debateinclude Gowers 2023, Franklin 2024, and Lange 2025.

A background issue in all of these debates concerns the extent towhich each particular non-deductive method plays anessentialrole in the justificatory practices of mathematics. This questionarises at both a local and global level. At the local level, aparticular piece of reasoning to justify a given result may beunavoidably non-deductive, yet the result may also be established bysome other, purely deductive piece of reasoning. At the global level,it may be that our only justification for certain mathematical claimsis non-deductive. The extent to which our use of non-deductive methodsis due to limitations in practice rather than limitations in principleremains an issue for further investigation.

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