Francis Bacon and mathematics: the critical tradition

More, perhaps, than any of his contemporaries, Francis Bacon has been the object of major criticism from all sectors of the nineteenth- and twentieth-century intelligentsia. Harvey's gibe concerning Bacon writing about natural philosophy in a Lord Chancellor's fashion was never really rejected. This judgement has been shared by a number of historians of philosophy who have preferred to assign Bacon's thought to the field of literature rather than to that of philosophy. Even amongst those who have acknowledged the philosophical nature of Bacon's works, many have proposed them as an example of vulgar utilitarianism.Footnote ¹ Amongst historians of science, Alexander Koyré concluded that Bacon failed to understand what was happening around him and that his role in the progress of science was ‘utterly negligible’; Jean Pelseneer went even further and called the Lord Verulam ‘an ignorant and impudent dilettante’.Footnote ²

More than anything, it seemed inexcusable to these critics that the alleged father of modern science should have denied the importance of mathematics and hypotheses in the fields of science and scientific reasoning. And, indeed, this view about Bacon's failure to recognize the role of mathematics appears to be supported by a profusion of passages throughout the Baconian corpus, which seemingly belittle mathematics. However, amongst the passages frequently quoted by his detractors, no reference is usually made to a paragraph in the essay ‘Of studies’ where we find that ‘[h]istories make men wise; poets witty; the mathematics subtile … So if a man's wit be wandering, let him study the mathematics; for in demonstrations, if his wit be called away never so little, he must begin again’.Footnote ³ A similar passage can be read in the Advancement, where mathematics is compared to tennis, which, although being ‘of no use in itself’, is ‘of great use’ for its ‘collateral and intervenient’ virtue of improving reflexes.Footnote ⁴

Interestingly, in reworking the Advancement into the De augmentis, Bacon moved this observation from the section devoted to mathematics to the Appendix t ardiva paedagogica, thereby making it clear that the role he envisioned for mathematical studies was principally pedagogic and propaedeutic.Footnote ⁵ As Descartes too was to declare in a few years’ time, thanks to mathematics it is possible to train the mind to eschew distractions and unorthodox reasoning, in the same way that one's constitution can be fortified through physical exercise.Footnote ⁶ According to Bacon, such medicine of the mind is requisite for human progress, both spiritual and material.Footnote ⁷ It is a condition of the correct understanding, and it is fundamental for both the interpretation of nature and the improvement of man's life. Such an exercise includes but is not limited to the Doctrina faber fortunae sive de ambitu vitae (The Doctrine Concerning the Improvement of One's Existential Conditions), and, in general, Bacon refers to it by the expression ‘georgickes of the mind’, or georgica animi.Footnote ⁸

In light of this, it is clear why Bacon should address his letter ‘Touching helps for the intellectual powers’ to Sir Henry Savile, one of the most prominent British mathematicians of the time, author of a successful commentary on Euclid, and founder of the chairs in astronomy and geometry at Oxford. The piece, which was originally intended as a part of the chapter about ars paedagogica in the De augmentis, contains perhaps the clearest definition of ‘georgics of the mind’; it does so in connection with the role of mathematical training for sharpening men's intellect.Footnote ⁹

These texts and editorial choices are particularly significant for Bacon's conception of mathematics as they implicitly call attention to the sixteenth-century humanist tradition of educational reform.Footnote ¹⁰ The influence of this tradition on Bacon was undoubtedly great, indeed greater than his antipathy towards figures such as Petrus Ramus would lead one to believe.Footnote ¹¹ Besides, the notion that ‘mathematics make men subtile’ also implies a reference to the early modern debate on the concept of subtilitas. In order to fully understand Bacon's stance on mathematics in the context of early seventeenth-century culture, it is necessary, therefore, to enquire into both these intellectual traditions, in order to show how their characteristic features were espoused, rejected or modified in Bacon's thought.

To this end, the second, third and fourth sections of this article will analyse the relationship between Bacon's thought and the humanist tradition concerning mathematics. The fifth and sixth sections will explore the concept of subtilitas, as it was put forward by Bacon and his sources. Upon these premises, in the seventh section I will propose a new reading of Bacon's understanding of mathematics suggesting that, although rejecting any form of mathematization of nature, he envisioned the possibility of using ‘naturalised mathematics’ for natural inquiry following the preliminary work of the Great Instauration.

Bacon and the topoi of the apology for mathematics

The humanist insistence on the need to found chairs in mathematics at the major European universities was accompanied by a set of apologetic arguments concerning the nobility and utility of mathematics.Footnote ¹² The practical use of mathematics for some professional categories, merchants in particular, had been fully recognized by the culture of the high Middle Ages.Footnote ¹³ However, the European culture of the sixteenth century extended the notion of practical mathematics to virtually all practical sciences and techniques, which were accordingly included in the quadrivium, and labelled mixed (applied) mathematics.Footnote ¹⁴ Humanists and reformers of the curriculum studiorum such as Jacques Lefèvre d'Etaples insisted that, rather than being limited to account-keeping and calculation of volumes, the utility of mathematics also encompassed a series of functions which had been previously fulfilled by logic.Footnote ¹⁵ The practical use of mathematics became a central feature in the works by Ramus and his mentor, Oronce Fine, whose Protomathesis of 1532 significantly concluded with an example of such practical use of mathematics; that is, a chapter on sundials, their nobility, and how to make them.Footnote ¹⁶ Yet the utility of mathematics was not merely practical. As Philipp Melanchthon put it in his Mathematicarum disciplinarum encomia, mathematics was necessary not only ‘for the “mechanics”, who need exact measurements in order to build homes, vases, and other artefacts’, but also for the philosopher.Footnote ¹⁷ The same principle was espoused in France by Ramus and Fine, who, just like Giorgio Valla and Lefèvre before them, considered mathematics to be ‘the key to all perfect knowledge’ since it ‘makes all secrets intelligible’.Footnote ¹⁸

Such notions concerning the generally propaedeutic function of mathematics were clearly favoured, especially in the French milieu, by the burgeoning attention devoted to authors such as Boethius and, in general, to the whole Platonic tradition, which had always held mathematics in high esteem.Footnote ¹⁹ Yet this was not an exclusive feature of the Platonically coloured French environment of Fine, Lefèvre and Ramus. On the contrary, the propaedeutic role of mathematics was commonly accepted across all sectors of sixteenth- and seventeenth-century European culture, from British mathematicians such as John Dee and Robert Recorde to members of the Jesuit order in Rome and in the many Jesuit colleges scattered throughout Europe.Footnote ²⁰

By the end of the sixteenth century the central arguments of the humanist defence of mathematics had crystallized into a series of topoi. The Baconian notion that the study of mathematics would aid prospective philosophers in the preliminary sharpening of their intellect was also amongst the chief points made by the humanists. In De disciplinis of 1531, the Spanish humanist Juan Luis Vives stated that arithmetic, which he deemed necessary to all aspects of human life, ‘puts the intellect to the proof, improves it, and makes it quicker’.Footnote ²¹ Through the study of mathematics, young people learnt not only a specific discipline, but also a rigorous method.Footnote ²² A few years later, the same pedagogical principles were endorsed by Ramus in his Scholae mathematicae.Footnote ²³ In Italy, Girolamo Cardano praised mathematics for its use ‘in cultivating the intellect, refining the other disciplines, and living a healthy and happy life’, while the Jesuit Christoph Clavius, in his Algebra, declared that no science could serve better than mathematics towards the goal of developing the subtlety of the intellect and the certitude of one's arguments.Footnote ²⁴

The immediate consequence of this conviction regarding the beneficial effect of mathematical studies on the intellect was that mathematics began to appear particularly necessary for the discipline that, amongst all, most required a process of emendation of the intellect: that is, philosophy. The motto ἀγεωμέτρητος ὀυδὲις ἐισίτω (‘Let none ignorant of geometry enter’), whereby Plato had allegedly forbidden access to his Academy to those who were unfamiliar with geometry, became the guiding principle for all the sixteenth- and seventeenth-century apologists for mathematics, from Vives to Fine and from Ramus to Melanchthon.Footnote ²⁵

Bacon's conviction concerning the usefulness of mathematics for sharpening men's wit, and in particular for training the intellect to understand complex problems such as those implied by natural philosophy, was certainly influenced by these arguments. However, while agreeing with the humanist reformers on the propaedeutic virtues of mathematics and on its practical utility for a variety of disciplines and practical sciences (such as glassmaking, pottery and ars militaris), Bacon differed from almost all of them with regard to the place of mathematics in the tree of knowledge.Footnote ²⁶ Quite originally, Bacon is convinced that mathematics should not be thought to occupy an autonomous position in the arbor scientiarum, but should rather be subsumed under some other more developed and hence autonomous science.

Mathematics, Bacon thinks, lacks axioms of a representative nature – that is, axioms which only apply to mathematics itself – in other words, it lacks a principium individuationis. It may be true that mathematics can be beneficial to all other disciplines; however, while its general laws may well be adopted by all sciences, they are not specific to mathematics exclusively and hence cannot characterize mathematics as an autonomous science.Footnote ²⁷ Even more significantly, once Bacon ruled out the possibility of granting mathematics, and especially mathematicae purae, an independent role in the tree of knowledge, he resolved to assign it to the field of metaphysics rather than physics. This choice is vindicated in the De augmentis by the fact that quantity – that is, the subject matter of mathematics – is, according to Bacon, the most abstractable from matter, and hence the most metaphysical, of all the formae essentiales. From Bacon's perspective, however, such abstractedness also implies the risk that mathematics may distract the natural philosopher from the study of things. Overly preoccupied with mathematics, the natural philosopher may thus yield to the inherent vice of the intellect, which, to the great detriment of sciences, is naturally inclined to favour the campi liberi of general abstractions over the silvae et septa of particulars.Footnote ²⁸

Medietas and abstraction in mathematics

The inherent tendency of mathematics to favour what Bacon regarded as a natural yet faulty predisposition of the intellect towards abstraction had also been taken into account by the humanist reformers of the curriculum studiorum, even though they mostly did not share Bacon's negative judgement in this regard. In the humanists’ works, the question of mathematical abstraction found expression in the debate concerning medietas mathematica; that is, the intermediate position of mathematical objects between sensible and intellectual entities (dealt with in physics and metaphysics respectively). The most authoritative source in this regard was probably Boethius’ De Trinitate, which compartmentalized speculative sciences into three groups: the ‘natural’ ones, which deal with motions and material (non-abstract) objects; the ‘theological’ ones, which do not deal with motions or with material objects; and the ‘mathematical’ ones, which occupy an intermediate position between ‘natural’ and ‘theological’ sciences. While mathematical sciences do not deal with motion, they consider forms which, although capable of abstract consideration, are in reality inseparable from material bodies.Footnote ²⁹ Thanks to its intermediate position between physics and metaphysics, mathematics can hence be regarded as a way of ascending (viandum) from one field to the other.Footnote ³⁰

This Boethian notion of mathematical medietas was embraced by virtually all humanist reformers. Lefèvre, who had published his first works in mathematics in the form of a commentary to Boethius’ De Arithmetica, was clearly much influenced by it. The same can be said, for instance, of Valla, who adopts Boethius’ stance on the three types of speculative science almost literally.Footnote ³¹ Fine, too, referred to mathematical medietas as one of the reasons for the nobility and utility of mathematics, which could be assimilated to the angelical spirit (esprit angelique), which is the link between God and the human world.Footnote ³² Fine understood that the intermediate position of mathematics between physics and metaphysics resulted not only in a propaedeutic advantage, but also in the possibility of upholding the logically necessary character of mathematical judgements.Footnote ³³

Indeed, the most significant implications of mathematical medietas were to be addressed in the sixteenth-century dispute on the certitude of mathematics. The official genesis of the debate can be traced to Piccolomini's Commentariolum de certitudine mathematicarum, published in 1547.Footnote ³⁴ Piccolomini's view, although encompassing a variety of considerations, hinged on the fact that in strict Aristotelian terms mathematics could not have been considered a bona fide science, since it did not produce demonstrationes potissimae; that is, the highest and most certain sort of demonstration described by Aristotle in the Posterior Analytics.Footnote ³⁵ Such a stance obviously provoked a cascade of critical responses coming from some of the best-known European humanists, such as Pietro Catena, Jacques Pelletier du Mans, and Cristophorus Clavius.Footnote ³⁶ Among these, Francesco Barozzi and his Quaetio de certitudine mathematicarum of 1560, which purported to show that mathematical demonstrations did in fact obey the Aristotelian standard of causal demonstration, played a particularly significant role.Footnote ³⁷

Medietas was the point at stake in both Piccolomini's and Barozzi's treatises. Opposing the principal objection to the scientific validity of mathematical proofs, which Piccolomini had formulated in his Commentariolus, Barozzi argued that even the theorems cited by Piccolomini as non-casual demonstrations (e.g. 1.32 of Euclid's Elements) could in fact be considered demonstrationes potissimae thanks to the specificity of mathematical middle terms.Footnote ³⁸ Medietas was hence the very quality that, accounting for the universal character of mathematics, was responsible for its epistemological primacy.Footnote ³⁹

The importance of mathematical medietas was also recognized by Piccolomini, who admitted that even though it was not enough to make mathematical demonstrations potissimae, it sufficed to grant mathematics a relative certainty resulting from the abstract nature of its object.Footnote ⁴⁰ However, at variance with Barozzi, Piccolomini showed little enthusiasm for abstractness and considered abstract mathematical objects inferior to material entities precisely because of their ‘intermediate’ and hence simplistic character.Footnote ⁴¹ Such a stance on Piccolomini's part contains a number of elements which were to be developed in Bacon's polemic against mathematics, alongside the originally Ramist notion that mathematicians were guilty of having yielded to the temptation of elitism and abstraction.Footnote ⁴²

Indeed, Bacon often seems to overturn some of Piccolomini's observations only to reach the same conclusions. Piccolomini, for instance, described quantity as ‘the most perceptible among perceptible things’ (omnium sensatorum sensatissimum) in an attempt to demonstrate that a science whose object was so gross had to be more simplistic than a discipline like natural philosophy, which studied qualities hidden ‘in the deep and obscure recesses of nature’. However, Bacon considered mathematics too simplistic if compared to natural philosophy, not because quantity was too gross a concept, but because, on the contrary, it was too abstract and detachable from matter.Footnote ⁴³ In the De augmentis, mathematics is defined as ‘the most abstracted and simple kind of knowledge’, and, for this reason, it is utterly incompatible with the study of intrinsically complex natural objects.Footnote ⁴⁴ Such incompatibility between mathematical method and nature is all the more evident considering that many natural objects are, in Bacon's words, ‘monodic’; that is, irregular. Nature thus requires a method of analysis that, unlike mathematics, may prevent the intellect from presupposing that there should be ‘a greater equality and uniformity than there is’.Footnote ⁴⁵ As he wrote in De principiis, Bacon found nothing to be more detrimental to natural inquiry than the habit of the philosophers who ‘have not taken the principles of things as they are found in nature’. Instead of accepting these ‘as a positive doctrine, resting on the faith of experience’, they had ‘turned … to the leges sermonum, conclusionunculae and communes notiones typical of mathematicians and dialecticians’.Footnote ⁴⁶

For Bacon, the mathematization of nature always involves a certain degree of abstraction from things and hence a tendency to think of nature as regular and uniform. On this premise, it is clear that all studies of nature based on general mathematical laws would have appeared utterly inadequate to Bacon, who made himself clear upon this point in addressing the issue of astronomy.Footnote ⁴⁷ In the wake of a debate that counted amongst its participants figures such as Ramus and Patrizi, Bacon observed that astronomers had become lost in a hypothetical realm.Footnote ⁴⁸ Though they had been dwelling for centuries on questions such as the ‘poles of rotation’, and ‘why they are fixed in such part of the heaven rather than in any other’, yet with their computations and hypotheses they could only demonstrate ‘how all these things may be ingeniously made out and disentangled’. They were utterly unable to explain ‘how they may truly subsist in nature’.Footnote ⁴⁹

The admonishment Bacon addressed to the astronomers of the time was, ironically, perfectly in line with Kepler's revolutionary works, which Bacon ignored.Footnote ⁵⁰ Astronomers should have accepted ‘spiral lines … and … contrarieties of motion’ and, moreover, they should have rejected the formositas illa mathematica, whereby all motions that were not circular and perfect had to be scorned.Footnote ⁵¹ These convictions were to find fuller expression in the Thema coeli, which Bacon devoted to the polemic against mathematical (or hypothetical) astronomy, and to the elucidation of a germinal astronomical theory, whose most innovative trait was the attempt to analyse astronomical matters ‘discarding logical [dialecticis] and mathematical subtleties’.Footnote ⁵² This, of course, would have led astronomers not only to admit of irregularities in planetary motions, but also to reject doctrines such as those about the stars being ‘parts of their own orb fixed as it were by a nail’. This, according to Bacon, was ‘very evidently a conceit of those who deal with mathematics not with nature, and fixing all their attention on the motion of bodies entirely forget their substances’.Footnote ⁵³

Mathematical syncretism

A further element that characterizes the mathematical debate from the fifteenth to seventeenth centuries is the syncretic attempt to demonstrate a substantial concordia philosophorum concerning the nobility and usefulness of mathematics. To present a picture of Plato as a strenuous defender of mathematical sciences was clearly quite straightforward, both due to his well-known motto and thanks to Boethius’ De arithmetica and Proclus’ commentary on Euclid, whose ultimate purpose seems to have been that of ‘Mettre les faits d'accord avec la philosophie de Platon’.Footnote ⁵⁴ The case of Aristotle might have posed greater difficulties, but the issues raised by the construction of an image of Aristotle as a champion of mathematics most often went completely unnoticed, or were disregarded altogether. On the contrary, sixteenth- and seventeenth-century works in mathematics can be considered some of the best specimens of integration of Platonism and Aristotelianism.Footnote ⁵⁵

The conviction that Plato's motto ἀγεωμέτρητος ὀυδὲις ἐισίτω was to be interpreted as an emblem of the whole Greek philosophy, instead of only applying to the Academy, became extremely widespread.Footnote ⁵⁶ In fact, according to Ramus and the humanist reformers, an adequate knowledge in mathematics was requisite for the right understanding of Aristotle's philosophy, even more than it was for Plato's. Johann Sturm, for instance, recommended that students in dialectics and Aristotelian logic attend preliminary courses in mathematics, in accordance with a notion concerning the affinity between mathematics and Aristotelian philosophy. This notion was also shared by Dasyposius, professor mathematicus at Sturm's gymnasium.Footnote ⁵⁷ Besides, a desire to show the necessity of mathematical studies to the comprehension of Aristotle was also common in traditionally Aristotelian milieux, such as that of the Jesuit order, whose Ratio studiorum, revised by Clavius, clearly stated that the Posterior Analytics ‘cannot be understood without mathematical examples’.Footnote ⁵⁸

The reference to Aristotle's use of mathematical examples was particularly significant because it also implied that mathematical knowledge was indispensable for the right interpretation not only of Aristotle's physics and logic, but also of his politics and ethics. Melanchthon, for instance, demonstrated that the Aristotelian theory concerning justice, as it was expressed in the fifth book of the Nicomachean Ethics, was to be understood ‘in a mathematical sense’; that is, in light of the distinction between geometrical and arithmetical proportion.Footnote ⁵⁹

Yet the debate regarding the compatibility between Aristotle and Plato on mathematical grounds found perhaps its most consequential outcomes in the context of the quaestio de certitudine mathematicarum, which also gave rise to a number of intransigent attempts to translate Euclidean mathematics in rigidly Aristotelian terms. Dasypodius’ Analyseis Geometricae sex librorum Euclidis are probably the best-known examples of such ‘extreme’ attempts to demonstrate the identity between mathematical propositions on the one hand and demonstrationes potissimae, understood as a type of argument which had to embody the properties that Aristotle ascribed to scientific syllogisms, on the other.Footnote ⁶⁰

According to Dasypodius, mathematical demonstrations are ‘constituted by many concatenated syllogisms’.Footnote ⁶¹ As a consequence, all theorems can be unfolded into a syllogistic chain.Footnote ⁶² However, as much as Dasypodius’ syllogisms were irreproachable from the formal point of view, they were also clearly superfluous for the outcome of the demonstrations they concerned, as many of Dasypodius’ detractors soon were to observe. Ramus himself, without referring to Dasypodius, considered Euclid's proofs to be self-evident: at most they called for explanation, but certainly not for demonstration, especially not for syllogistic demonstration, which merely obscured already complex pieces of reasoning.Footnote ⁶³ A similar conviction was widely shared in the sixteenth and seventeenth centuries: in his Pathway to Knowledge, Robert Recorde reproduced Euclid's propositions, omitting their proofs for the sake of simplicity. Some fifty years later, Savile criticized Dasypodius for his abstruseness.Footnote ⁶⁴ It was clear that, although the mathematical method could at least in part be translated into syllogistic form, such an operation was ultimately futile, in particular when the entire syllogistic chain was made explicit, as in Dasypodius’ case.Footnote ⁶⁵

Bacon's polemic against mathematics can be interpreted based on the intellectual background that produced works such as that of Dasypodius. The tendency to consider mathematics and Aristotelianism as compatible fields provided the basis for the fundamental critique of the mathematical method expressed in the Novum organum, where the possibility of translating mathematical demonstrations into syllogistic chains was implicitly accepted. Bacon even devised a few examples of conformity between Euclidean postulates and Aristotelian syllogisms. For instance, he noted that the law whereby if ‘two things are equal to the same thing they are equal to one another … is conformable with the rule of the syllogism in logic which unites propositions agreeing in a middle term’.Footnote ⁶⁶ It is, however, evident that Bacon's position was not neutral. Much less did he desire to engage directly with the treatises on certitudo mathematica, whose themes nonetheless served as a source of inspiration to him. Bacon's criticism ultimately depended instead on his theory regarding syllogisms, which, projected onto the mathematical sciences, implied the inadequacy of mathematics or, at least, of a certain type of mathematics, for natural enquiry. If mathematical demonstration could be equated with syllogistic reasoning, then Bacon's critiques of syllogism would prove relevant to the fields of mathematics as well:

Subtilitas (or subtilitates) in Girolamo Cardano and Julius Caesar Scaliger

As much as the hitherto outlined Baconian view on mathematics might seem rather negative, I do not think it should be concluded, as it frequently is, that according to Bacon mathematical sciences could only prove detrimental to natural enquiry, displaying at the most some pedagogic qualities resulting from the fact that they make men subtile. Indeed, the assumption that mathematical studies might improve the subtlety of the intellect not only alluded to the humanist reform of the curriculum studiorum, and to the sixteenth-century literature on the nobility of mathematics, but also implied both Bacon's theorizations on subtlety, and the sixteenth-century debate on the same subject, which finds its archetypal illustration in two texts that Bacon often quotes: Cardano's book De subtilitate, and Scaliger's polemic response, the Exercitationes de subtilitate.

Both written in the 1550s, these texts were fundamental in normalizing one of the most nuanced concepts in the history of ideas. While the notion of subtilitas had been used with regard to matter from late antiquity to the age of Bacon in reference to roughly the same physical qualities (impalpability, imperceptibility, incorporeality or quasi-incorporeality), the definition of rhetorical or intellectual subtilitas had changed significantly across the centuries. For Tertullian, subtilitas in thought and speech is the almost exclusive mark of heresy.Footnote ⁶⁸ Argutiae et subtilitates are the means whereby subtilissimi haeretici sophistically call into question the received truths of faith by devious argumentation and lexical ambiguities.Footnote ⁶⁹ Yet rhetorical subtilitas was soon to be turned into a positive quality. A couple of centuries after Tertullian, Augustine is still uncertain: he conceives of subtlety as the attribute of sophisticated but orthodox argumentation, yet also sees a risk in over-sophistication.Footnote ⁷⁰ By the time Aquinas wrote, subtilitas had become an entirely positive intellectual virtue – so much so that subtilissime et efficaciter are the adverbs that Aquinas uses to describe the words of the Lord (Exodus 3:6) which Jesus cites to convince the Sadducees of the resurrection (Matthew 22:32).Footnote ⁷¹

All these definitions of subtlety, positive and negative, are reworked by Cardano in his notion of subtilitas, which also incorporates the originally Boethian idea of a special connection between mathematics and subtlety.Footnote ⁷² In light of all the past conceptions of subtlety, Cardano distinguishes between a subjective (or intellectual) type of subtilitas, and an objective (or physical) one. If arithmetic and geometry belong to the field of subjective or intellectual subtlety, since they concern especially complex topics, equally subtle are also some natural properties which Cardano refers to as secreta naturae. These secrets of nature are subtle not only because they are not easily comprehensible, but in particular because they are intrinsically unsubstantial (tenuis), be it due to their impalpability (air), uncountability (hair), continuous flux (blood), infinitesimal divisibility (gold), or combination of some of these properties (fire).Footnote ⁷³

Complex or abstruse notions, and nearly imperceptible or insensible natural objects, can thus be held equally subtle, although in a different manner, the one conceptual and the other material. As a consequence, it is also possible to postulate a third type of subtilitas, which consists in the quality of the intellect that can readily penetrate subjective or objective subtlety.Footnote ⁷⁴ Unlike the other kinds of subtilitas, this third type displays an artificial or human character and is exemplified by certain achievements of the arts. Such human subtlety could be defined as the conception of a subtle mind; that is, of a mind that is capable of penetrating natural and intellectual subtleties, being thereby in the position to operate in a way similar to that in which nature does. Human subtlety thus results in the artificial reproduction of natural processes, and its fruits must be placed in the field spanned between natural magic on the one hand and experimental science on the other. It includes, for instance, the ability to make colours transmute into one another, to turn wine into vinegar, or to invent flying machines and fire-resistant materials.

This notion of subtilitas, which proved quite influential for Bacon, was to be harshly criticized by Scaliger. As a matter of fact, all the specific objections arrayed by Scaliger against the De subtilitate reflect a fundamental dissent concerning Cardano's definition of subtlety:

The extent to which these objections influenced Bacon's reading of Cardano's De subtilitate is certainly not great; however, Scaliger undoubtedly played an important role in better defining Cardano's various distinctions amongst the types of subtilitas, which in their original formulation appeared quite nebulous. The fundamental critique expressed in the Exercitationes concerns the impossibility of describing as subtilis everything that comes near to the limit of perceptibility or intelligibility. According to Scaliger, Cardano's notion of subtlety not only was extrinsic to nature, concerning only the human mind, but also lacked coherence, from both the physical and the logical point of view. In other words, it owed its ontological status only to Cardano's intellectual flaws.

These critiques also applied to the fourth kind of subtlety theorized by Cardano in the fifteenth book of the De subtilitate: useless subtlety. A definition of subtilitas inutilis is not provided; however, it can be deduced from the examples chosen by Cardano, such as the ‘baguenaudier’, also known as the Chinese rings puzzle, which he defines as a ‘game devised for its admirable subtlety’.Footnote ⁷⁶ Inutility and subtlety also characterize intellectual games such as the cross-shaped compositions by Rabanus Maurus, which, albeit marvellous and elaborate, are useless since they are not based on the three pillars of writing: ‘present use, clear purpose, and solid premises’.Footnote ⁷⁷ Useless subtlety becomes deplorable in the case of Ramon Llull, who ‘wanted to divulge all sciences, yet knew none’, and promised to teach all things, only to waste his time in the practice of a combinatory art which Cardano – like Bacon – holds detestable.Footnote ⁷⁸

Yet subtilitas inutilis was not a Cardanian invention. The same concept had appeared in the works of Montaigne, one of Bacon's prime sources. The fifty-seventh chapter of the first book of the Essais was dedicated to the vainez subtilitez (‘vain subtleties’), and shared with the fifteenth book of the De subtilitate not only the subject, but also some of its material and examples. However, perhaps Montaigne's most significant influence over Bacon concerns his bias against useless subtleties altogether. Unlike Cardano, who criticized the deceitful ones, but considered other cases innocuous, if not pleasant, Montaigne made no distinction among subtilitates inutiles. In the Essais all useless subtleties are deemed reproachable since they originate from the vice of excessive intellectualism, which is supremely exemplified, both for Montaigne and for Bacon, by Scholastic philosophy:

In addition to this, Bacon's cultural environment was largely influenced by the Elizabethan culture, whose opinion on subtlety was extremely negative, as proven by the works of many sixteenth-century English didactic writers. Subtlety, useless and deleterious by definition, was the specific mark of Italian Machiavellians (Iago/Volpone), and in general of the kind of people that good and simple Englishmen had to flee.Footnote ⁸⁰ As Roger Ascham wrote in the Schoolmaster, perhaps the most important Elizabethan pedagogical treatise,

Subtilitas in Francis Bacon

Bacon's notion of subtilitas, so important a concept as to be cited twenty-nine times in the Novum organum alone, had much in common with all hitherto cited doctrines.Footnote ⁸² Not unlike Cardano, Bacon distinguished various types of subtilitas; furthermore, like Scaliger, he paid particular attention to what could be defined as subjective or intellectual subtlety; and, like Montaigne and Ascham, he underlined the immoral character of all types of useless subtlety. Yet Bacon did not embrace any of these views in a rigorous manner, developing instead a radically new perspective, which must be considered with reference to the role of mathematics in the Great Instauration.

Bacon, like Cardano, believed that some aspects of nature were inherently subtle. This conviction is clearly implied by the classic metaphor whereby Bacon described the universe as a forest (sylva), and the natural philosopher as a hunter, intent upon following a trail of nearly imperceptible traces. Such a method of enquiry was founded upon the assumption that deductive (causal) knowledge is impossible to attain since the causae rerum are too remote and subtle to be comprehensible a priori and with clarity.Footnote ⁸³ The very notions of processus and schematismi latentes (‘latent processes’ and ‘latent schematisms’), whereby Bacon refers respectively to the hidden formal and material causes of things, are an example of natural subtlety. What is more, Bacon often refers to some subtiliores schematismi (‘more subtile schematisms’), which are understood to represent nature's most arcane structure ‘in which the occult virtues and sympathies of things chiefly reside’.Footnote ⁸⁴ These subtiliores schematismi are of paramount importance for the natural philosopher, because – as Bacon asserted in many passages of his works – the subtlest aspects of things provide the most accessible path to the right understanding of nature. They are ‘another key to open nature, as well as the sense; and sometimes better’.Footnote ⁸⁵ Indeed, ‘that secret of nature, of the turning of iron touched with the loadstone towards the north, was found out in needles of iron, not in bars of iron’.Footnote ⁸⁶

In light of nature's exceptional subtlety, Bacon praised Democritus’ atomism above all previous philosophical traditions. Although Bacon never adhered to such a philosophical stance completely, he believed that

On the contrary, the Baconian philosopher had to reject decisively all philosophies that could be considered uselessly subtle. To this category belonged not only Lullism, but all the doctrines which Bacon condemned for having dealt with verba (words) rather than res (things). As in Montaigne, this attack was not directed at intellectual subtlety per se, but rather at an excessive intellectualism resulting in abstraction from things, and confining debates to a self-referential sphere built upon meaningless words.

As Cardano had done before him, Bacon gives as an example of useless subtlety Proclus’ commentary on Euclid.Footnote ⁸⁸ In it, claims Bacon, the author endlessly indulges in obscure speculations that are demonstrative of the typical vice of the intellect, which ‘when … [it] cannot master the solid … wastes itself on the superfluous’.Footnote ⁸⁹ The reproachable category of useless subtleties thus encompasses all pure mathematical abstractions, which Bacon equated to syllogisms inasmuch as both are utterly removed from things. Indeed, Peripatetic and Scholastic doctrines are the principal target of Bacon's critique. They are the example par excellence of subtilitas inutilis, as evidenced by the use of syllogistic methods aimed at ‘coercing’ the opponents’ arguments thanks to some ‘subtlety of words’ instead of essaying for the truth, through natural enquiry.Footnote ⁹⁰

‘[P]hilosophies dealing with troublesome subtleties, dogmatic, weighing and wresting everything to the standard of certain principles; and lastly, those that were crabbed and narrow, were bad; and such were mostly the sects of the peripatetics and schoolmen’.Footnote ⁹¹ Peripatetic and Scholastic doctrines, together with mathematical abstractions – Bacon believes – are not different from Cardano's intellectual games and from all the frivolities which have nothing to do with things but only concern words. The threat posed by the deplorable philosophy of verba is always present, and even the most solid doctrines can be infected with forms of ‘quarrelsome subtlety’ (litigiosa subtilitas). Just as organic bodies rot and end up eaten by worms, so ‘it is the property of good and sound knowledge to putrefy and dissolve into a number of subtile, idle, unwholesome, and (as I may term them) vermiculate questions’.Footnote ⁹²

In light of this, two principal modes of conceiving the notion of subtilitas can be found in Bacon. On the one hand there is natural subtlety, which is inherent to the structure of things and therefore cannot be made an object of moral judgement; and on the other, there is human subtlety, which concerns the operations of the intellect and becomes subtilitas intutilis when it results in verbose and abstract doctrines, such as Scholasticism, to the corruption of natural philosophy. To these types of subtlety must be added, however, a third kind of subtilitas, previously considered by Cardano. It is the specific kind of human subtlety which, instead of marking the intellect's faults in preferring verba over res, marks the intellect's ability to penetrate natural subtlety, as required by natural philosophy.

The Baconian enquirer must therefore educate himself through the georgics of the mind, which also include mathematics, in order to improve his intellect's penetrative capacity. Once edified in a similar way, the intellect of the natural philosopher will convey a noble kind of subtlety (subtilitas intellectus). The Baconian philosopher, thanks to his self-discipline, will only exercise this subtlety upon things, leaving aside the sterility of words. Moreover, just as in Cardano, for Bacon this kind of subtilitas is evidenced by the devising of ingenious artefacts aimed at emulating natural processes, such as clocks, modelled after the ‘circular motion of the celestial bodies and the continuous and orderly movement of the animal pulse’.Footnote ⁹³ However, while in Cardano artificial subtlety (or subtilitas intellectus) seemed to tend towards natural magic – that is, towards an attempt to influence nature and redirect its effects through the imitation of its means – in Bacon, subtlety of the intellect inclined instead towards the opposite pole, that of experimental science.Footnote ⁹⁴ Subtilis is thus the adjective which Bacon customarily uses to describe a particularly fruitful and keen experiment; that is, an experiment which can make particularly evident natural causes by their effects.

To sum up, Bacon incorporates three kinds of subtlety into his system of thought, all of which find their model in the sixteenth-century traditions of subtilitas. In the first place, he postulates the existence of a subtilitas rerum, intrinsic to things; in the second place, he criticizes subtilitas verborum, which, being connected to syllogistic logic and to pure mathematical abstractions, is always detrimental to natural philosophy; and in the third place, he admits a kind of noble subtlety, subtilitas intellectus, which is exercised upon the subtilitas rerum and can be defined as the intellect's penetrative virtue that is necessary for the production of experiments.

In various passages of his works, Bacon compares these types of subtilitas with one another. In the Valerius Terminus, for instance, we read that

In other words, since the subtilitas rerum is more subtle than the subtilitas of the senses, the paramount risk for the natural philosopher is that of averting his eyes from the forest of things, giving way instead to wordy speculationes – that is to say, to the subtilitas verborum.

However, even subtilitas verborum cannot parallel the subtlety of things, as clearly appears from the fact that syllogism, its principal fruit, ‘commands assent … to the proposition, but does not take hold of the thing’.Footnote ⁹⁶ Indeed it is impossible ‘that axioms established by argumentation should avail for the discovery of new works; since the subtlety of nature is greater many times over than the subtlety of argument’.Footnote ⁹⁷ The only means for the philosopher to grasp the subtlety of things is hence provided by the subtilitas intellectus. Since the senses per se are not sufficiently subtle to comprehend the complexity of reality, one should appeal to what we may call a ‘prosthetic sense’ that could, at least in part, cover the ground between the subtilitas rerum and the naked senses. Therefore subtilitas intellectus avails the natural philosopher not only by increasing the acuteness of his intellect, but also by providing him with such a ‘prosthetic sense’, which is the ability to produce experiments. Along these lines, experiments can be connected to the study of mathematics since they are the chief product of noble subtilitas intellectus, which is exactly the kind of subtlety that Bacon is referring to in stating that ‘mathematics make men subtile’.

As much as these observations have often been ignored by Bacon scholars – perhaps due to the fact that Bacon himself never distinguished in a clear fashion between the various types of subtlety he contemplated – the relationship he established between the three kinds of subtilitas is of paramount importance to the understanding of the role of mathematics in the philosophical system of the Great Instauration. A certain abstract use of mathematical methods in the field of natural philosophy might indeed be equated by Bacon to the use of syllogisms, thus pertaining to the province of execrable subtilitas verborum. Yet at the same time it must be acknowledged that mathematics, if used correctly (that is, if directed to the study of things), increases the philosopher's subtilitas intellectus within Bacon's system. A passage of the De augmentis is extremely clear in this respect. Upon considering the topic of mixed and pure mathematics, Bacon conceded that ‘many parts of nature can neither be invented with sufficient subtlety, nor demonstrated with sufficient perspicuity, nor accommodated to use with sufficient dexterity, without the aid and intervention of Mathematics’.Footnote ⁹⁸ In other words, some of the subtle recesses of nature can be comprehended only through the careful use of mathematics. Bacon admits this principally in one case: when genuine mathematical procedures are directed to the analysis of things in an utterly non-abstract way.

It was therefore fundamental for the natural philosopher to carefully distinguish between subtilitas intellectus and subtilitas verborum and, at the same time, between a science which could be named mathematica rerum and its opposite, mathematica verborum. According to Bacon, abstraction is always reproachable, meaning that a priori speculations such as those of the Aristotelian philosophers and pure mathematicians cannot but contrast with the ultimate purpose of natural philosophy. Yet once the Baconian philosopher has collected a sufficient amount of natural data, he must not overzealously scorn an exercise of subtilitas intellectus which can help in grasping the unity underlying natural data, so as to arrive at the formulation of axioms; that is, of general laws whose aim is to unify natural particularity:

This aspect of the inductive method clearly has nothing to do with the subtlety of words, which always tends towards the construction of a priori systems and schemes; on the contrary, it aims at the definition of a unified organon based on things, and hence a posteriori.

Conclusion: prospective science and mathematica rerum

Mathematics can hence play an important role in the context of Bacon's natural philosophy; however, in order for this to happen, a clear distinction must be drawn between the aspects of mathematics which pertain to subtilitas inutilis (verborum), and those that instead belong to the field of subtilitas intellectus. In other words, the natural philosopher needs to discriminate between useless a priori abstractions and a posteriori mathematical enquiries carried out on things. The use of ‘real’ mathematics – that is, of a kind of mathematics that, in spite of its medietas, is anchored to the study of things – remains somewhat utopian in Bacon's thought. As Guido Giglioni has observed, once the original contact between res and verba is recovered through the Great Instauration, it will be possible for the Baconian philosopher to substitute the mathematical object for the physical (material) one.Footnote ¹⁰⁰

This does not mean, however, that Bacon should imagine prospective science, heir to the Great Instauration, to have something in common with the views that were to be proposed by Galileo and, later, by Newton. According to Bacon, it will always be impossible to found natural philosophy on mathematics; yet the time may come, on the contrary, when mathematics could be founded upon natural philosophy, in order to retain only the best of what mathematics can offer.Footnote ¹⁰¹ What is hence at stake is not the mathematization of nature, but, on the contrary, the ‘naturalization of mathematics’, which is to say, the foundation of a mathematica rerum.

In light of this, one can fully appreciate the meaning of a passage from the Novum organum:

According to Bacon's constitutionally forward-looking and progressive conception of science, it is possible to think that, after the Great Instauration, and after the foundation of the ‘mathematics of things’, perhaps the very method of induction could be replaced by a kind of mathematica rerum. In Bacon's view this would be radically different from Galilean, Cartesian, or Newtonian science, since it would not be based upon abstraction. The allusion to the one, the thousand, and the thousandth part of an integer may then be seen to imply the Baconian notion of perfect induction. But the very reason why induction needs to be perfect is that the Great Instauration, at the time of Bacon's writing, had not yet taken place, nor had the contact been recovered between the intellect and the things of nature. Only following the Great Instauration will it be possible to equate one with one thousand – that is, to admit the principle of representative examples, which is at the core of imperfect induction and so at the heart of a truly ‘modern’, Newtonian science.

Provocatively, one could state that through the mathematica rerum wished for in the Novum organum, Bacon might have somehow foreshadowed the possibility of ‘getting from Bacon to Newton’, even though necessarily this could only occur once one had absorbed and surpassed the Great Instauration – something that Newton, clearly, did not do.