Bernard Bolzano (1781–1848), known in his lifetime as ‘the Bohemian Leibniz’, is best known as a logician and mathematician, but he also developed a monadology in which the monads, which he called ‘atoms’, have spatial location and physical properties. This essay summarizes and assesses his monadology.
Bernard Bolzano (1781–1848) is best known for the throroughgoing and astonishingly modern semantic reform of logic proposed in his monumental Wissenschaftslehre (1837). While little known or regarded in his time or for a long period thereafter, this has gradually come to be recognized as one of the most substantial contributions to the subject between Leibniz and Frege. He is also known as an innovative mathematician, an early proponent of clear, purely mathematical conceptual foundations for the subject, an enthusiastic supporter of the actual infinite, and a forerunner of Cantor in set theory. Much less known is Bolzano's metaphysics, which is a thoroughgoing monadology, proposing the existence of an infinity of partless substances, physical and mental, completely filling space, and interacting via forces of attraction and repulsion. Bolzano closely studied the critical philosophy of Kant, and disagreed with it on many points. His advocacy of the actual infinite and his confidence in the idea of a realist cosmology of simple substances represented a qualified return to Leibniz, whose work he admired greatly, but with aspects of which he also disagreed. Like Leibniz, Bolzano had polymathic talents – not just logic and mathematics, but also theology, ethics, aesthetics and physics were all enriched by his work – and he was known even in his own times as ‘the Bohemian Leibniz’. Bolzano was one of the nineteenth century's clearest and most systematic thinkers, so his metaphysics deserves to be better known: this essay attempts a critical summary. Yet his views particularly as regards their import for physics are also irremediably archaic, coming just before the electromagnetic revolution of Øhrsted, Ampère, Faraday and Maxwell, remaining wholly descriptive rather than quantitative in form, and in Bolzano's striving for apriority, beyond verification or falsification. The failure of a mechanical substance–attribute monadology of even such an astute thinker as Bolzano must raise doubts as to whether any such account is viable.
Bolzano's mature metaphysics is found in three main sources, given here in chronological order. The first is his large monograph Athanasia, first published anonymously in 1827, then in an enlarged and improved second edition in 1838 under the author's name. The only work of Bolzano's to achieve any popularity in his lifetime, it is, as the title says, a defence of personal immortality. Its full title in approximate English translation is Athanasia, or Reasons for the Immortality of the Soul. A Book for Every Educated Person Who Wishes to be Reassured about This. The second source is Bolzano's ‘Aphorismen zur Physik’ (‘Aphorisms on Physics’), dating from 1840, read and discussed in 1841 at the Royal Bohemian Society of Sciences, but never put into a form he deemed worthy of publication. It was edited and published in the Gesamtausgabe (Complete Edition) by Jan Berg in 1978. The final major source is the second half of Paradoxien des Unendlichen (Paradoxes of the Infinite) of 1851, edited by and published posthumously in 1851 by Bolzano's pupil Franz Příhonský. Bolzano worked on the Paradoxien in 1846–7. Its title is apt to mislead. It is in fact a sustained defence of the notion of infinity against claims that it is paradoxical, absurd or self-contradictory. The first part of the book deals with infinity in mathematics, but the second part, which is what interests us here, deals with infinity in physics, or in the physical world. A more compact but less complete source of information on Bolzano's monadology is an essay, ‘Atomenlehre des sel. Bolzano’ (‘Atom Theory of the late Bolzano’) by Příhonský. While Bolzano did add a little to his views over time, he revised relatively little, so we can treat his monadology as a theoretical unit and do not need to distinguish different phases of development.1
As argued in Athanasia and presupposed elsewhere in Bolzano's writings, he accepts that among real things (wirkliche Dinge), those which are to be found in space and time, and enter into causal relations, there are two fundamental sorts: substances and adherences. A substance is an ontologically self-sufficient thing that ‘exists for itself’ (AA 21), is a non-predicable (AA 1) individual (WL 142, PdU 57), as in the tradition from Aristotle to Leibniz. Adherences are the individual characteristics of substances and of other adherences (AA 9). Adherences are of or in other things (an etwas Anderem), substances are not – for example the matter of which a body is composed is substance (AA 21). Adherences are particulars which depend on their substances for their existence and which characterize or qualify these substances, and can be called their properties (Eigenschaften) (WL 272.2). For example the colour, weight and smell of a body are adherences of it (AA 21). Although not always quite consistent in his usage, he tends to call the kinds or types of adherences ‘attributes’ (Beschaffenheiten), but like Leibniz he thinks the real things, things having existence (Daseyn) are all particular (AA 9). Though no nominalist – unlike Leibniz – Bolzano does not make great play with the notion of universal or general attributes in regard to real things. He does allow that non-real things such as concepts and numbers have attributes (AA 22), and cannot have adherences, since the latter are real. For present purposes it is enough to say there are substances, and how they are is determined by what adherences they have. The spatial location of a substance – since for Bolzano, unlike Leibniz, all substances are in space and have physical characteristics – is not an attribute or adherence but what Bolzano calls a determination (Bestimmung) (cf. Morscher, Das logische, 69 ff.). Likewise, the time at which a substance has a particular adherence is a determination of the substance: that which has the attribute ‘is sitting’ because of the existence of an adherence of Socrates's sitting is not Socrates simpliciter but Socrates at such and such a time. ‘[B]y the word “time”’, writes Bolzano, ‘we mean nothing but that particular determination of an actual thing which is the condition for ascribing truly a certain attribute to that thing’ (WL 79). Neither space nor time is real, since they are neither substances nor adherences (PdU 17), nor do they have effects (WL 79).
Bolzano has an argument for the existence of substances (AA 22) which is a close analogue of his cosmological argument for the existence of God. It goes as follows.
Something real exists.
Everything real is either a substance or an adherence.
There either are substances or there are not.
If there are not, then everything real is an adherence.
But adherences are all dependent entities.
Therefore the totality (complete collection) of adherences is also dependent.
It must then depend on something real that is not an adherence.
This must then be a substance.
Therefore at least one substance exists.
The flaw in this argument – leaving the classificatory presumption 2 aside – is step 6. Why could not all adherences be dependent – that is, each be dependent on something – yet groups of them which depend mutually on one another not constitute substances? The analogous flaw is present in the cosmological argument (Ganthaler and Simons, ‘Bernard Bolzanos kosmologischer’).
Bolzano notes that since we know we have ever changing ideas and judgements, since these changes are real, and we can be certain that they must be adherences, their substances are the bearers of change, so that the mental and physical properties and events that affect us are our adherences, but we ourselves are substances (AA 25).
Bolzano makes a fundamental distinction among real things between those which do and those which do not have (proper) parts. Real things with parts are composed (zusammengesetzt) of these parts, and Bolzano calls them collections or aggregates (Inbegriffe). A collection is simply anything which is composite, that is, which has (proper) parts. Bolzano does not make a clear distinction between collections as pluralities of individuals (e.g. the collection of books in my study) and collections as composite individuals (e.g. the single clock in my study), and it is futile to look for such a distinction – later made much clearer by such writers as Russell and Leśniewski – in his work. All composite objects (collections) are made of simpler objects, and Bolzano holds that it is obvious they are all ultimately composed of objects which are utterly simple, so lacking parts. These correspond in their simplicity and fundamentality to Leibniz's monads, but Bolzano disagrees with Leibniz about the relationship of his simples to space and to one another. Whereas Leibniz's monads are strictly not in space and have no spatial characteristics, being all of the type of indivisible mental substances or souls, Bolzano, in keeping with common sense and the physics of his day, takes his atoms to have a spatial location or determination, which varies when the atom moves. Since atoms have no parts, they cannot spread out over a volume and therefore at any one time the location of an atom is a spatial point (AA 117, AP 4). Because atoms have no extension, they have never been perceived (PdU 50), but despite this, it is ‘laughable’ to suppose for this reason that they do not exist (ibid.) Bolzano holds that the crucial assumption leading Leibniz to his idealistic or panpsychistic conception of monads is the principle of pre-established harmony, which arises out of Leibniz's attempt to overcome the mind–body problem and to avoid the assumption of transeunt causation. Bolzano thinks he can happily and successfully explain both mind and body through the forces of attraction among his atoms, and so need not deny that there are causal interactions among atoms and so between mind and body. Bolzano's theory of simples is thus, like that of the pre-critical Kant, also a physical monadology, the physical and the mental being co-comprised among real substances and adherences. He seems not to have known or at least cited this early Kant work, but does know the not dissimilar work of Boscovich, whose forces Bolzano claims to be substances rather than adherences (PdU 57), whereas he takes his own adherences to be forces (PdU 57) or mental phenomena (WL 143), whose bearers are substances, ultimately atoms.
Like Leibniz, Bolzano argues that atoms cannot be created or destroyed by natural means (AA 69), since the only way in which natural processes can create or destroy is by the rearrangement or reconfiguration of atoms, by their coming together and dispersing again. No atom has a beginning in time (AA 74) or an end (RW 214–27). Thus in the Athanasia, which has a deliberately consolatory message, and was written with Bolzano's friend Anna Hoffmann in mind, the death of whose daughter had first brought them into contact, those atoms which are human souls are likewise indestructible, and so exist forever (AA 68–84). Whether such persistence endows or re-endows them with the kind of conscious mental characteristics they enjoy during life is another matter, answerable less by natural philosophy than by religious faith. That there are atoms and among them souls or minds is taken by Bolzano to be evidenced by introspection, of mental phenomena together with the recognition that these themselves are adherences (AA 16, 25). God's creation of atoms, including ourselves, cannot therefore be creation by bringing into existence at a point in time, but must be the holistic, non-temporal creation of the atom as an object existing throughout time (AA 79).
The adherences of atoms are forces that act on other atoms (PdU 57). The forces can be attractive (AA 48) or repulsive (AP 19, PdU 63–4). (The thought that they might not act along the line between the atoms, as in electromagnetic induction, appears not to have occurred to Bolzano.) The German word Kraft that Bolzano uses here can mean both ‘force’ and ‘power’, and so covers both the disposition to act and the acting itelf. There is mutual interaction between atoms through their adherences. In fact, every atom permanently and continuously influences every other (PdU 60), so there is instantaneous action at a distance (AA 67), but no interpenetration: no point can be occupied by more than one atom (PdU 54). All atoms are changing all the time (PdU 50), and there is no ‘dead’ or purely inert matter (PdU 51). The intensity of the interaction or mutual influence (einwirken) varies considerably; however, as not all atoms have the same power (AA 21), they move relatively to one another, and the density of material objects varies and is not uniform (AA 16). Bolzano's is a through and through causal monadology, unlike that of Leibniz. He readily accepts a Newtonian inverse square law for the mutual attraction of atoms (AA 27), but not at all distances. Noting that solid bodies are hard to compress, he postulates, in ideas not dissimilar to those of Boscovich, that for any two atoms there will be a distance below which they mutually repel, and a distance above which they mutually attract (AP 63, PdU 63). The distance g below which mutual repulsion operates may be smaller than the distance G above which attraction operates. Distances between endow the atoms with a ‘comfort zone’ in which neither attracts nor repels the other but as it were ‘happy’ or ‘comfortable’ with them: Bolzano uses the German words genehm (‘acceptable’) and wohltuend (‘beneficial’). When an atom has another one closer to it than g, it is ‘uncomfortable’, meaning it will endeavour (bestreben) to repel the other, whereas atoms further away from it than G it ‘prefers’ to have closer and so will attract them (PdU 63). The size of a substance's comfort zone may vary with time (AA 21). Despite their standard psychological meanings, the words in Bolzano's usage are purely mechanistic in import. When two atoms are at a comfortable distance, that simply means they are disposed neither to attract nor to repel one another. This can be illustrated by the way in which magnetic levitation repulsive magnetic forces balance attractive gravitational forces and the bodies remain in equilibrium and at rest (not Bolzano's example).
Among atoms, some exercise a greater influence on those around them than any of those around them exercise on them in return. Bolzano calls such atoms dominant (herrschend) or distinguished (ausgezeichnet). A dominant atom influences the infinitely many atoms around it that have less power, though no dominant atom has infinitely more power than other atoms (PdU 62). The idea, though not its causal interpretation, is already present in Leibniz. Bolzano goes so far as to identify dominant substances as mental and dominated substances as material, though he admits that the difference is one of degree and not sharp. By taking the mental and the physical both to consist of atoms of variable dominance, he takes himself to have solved the mind–body problem: since the soul is an atom (indeed a dominant one) and the body consists of many atoms, there is no mystery about their interaction, which in some cases is immediate (unmittelbar) (AA 64, PdU 56). The soul, a spiritual (geistige) substance, is in space just as much as is the body (PdU 55). Body and soul develop together, and human reproduction shows that they are of the same kind (gleichartig) (AA 59). That Bolzano can hardly be taken to have solved the mind–body problem in such a way is not at issue, but there is no doubting the directness and boldness of his proposal.
Every atom is at any time at a point in space (AA 4) and a point cannot host more than one atom (AA 17). Bolzano's physical world is a plenum: there is no space that is not completely filled by atoms, not even a single point. But motion is possible even in a plenum (AA 15), because despite completely filling space, the atoms in it can be compressed together without any of them coinciding and expanded from one another without leaving holes. To motivate this thought, Bolzano notes, using a now-familiar argument, that there are just as many points on a line of given length as there are on a line of greater or lesser length (AA 17, PdU 20). The variability of density of the aether and other occupants of space allows for the transmission of waves by compression, rarefaction and lateral motion, about which Bolzano would latterly have been well informed through his friendship with Christian Doppler.
Most of space is empty of bulky matter. The balance is made up by atoms of aether, which is a pervasive, space-filling fluid (AP 25) or world-stuff (Weltstoff). Evidence for its existence comes from the wave theory of light, delays of comets and the shape of comet tails. Radiant heat passes through the aether as a transmissive medium (AP 52). Its existence is unverifiable because it pervades all matter so its mass cannot be determined (AP 27). Loss of aether mass via radiation also cannot be tested because mass and volume are not in a fixed proportion: density varies (AP 16). Every atom attracts aether atoms, but atoms which attract other than just aether atoms are distinguished (AP 29). Aether atoms attract only one another so are the least dominant or most dominated of all; distinguished atoms are all at a level above aether atoms, but distinguished atoms also come in different kinds with different degrees of dominance.
Material bodies are finite in volume whereas atoms have zero volume. Therefore, every material body is made up of infinitely many atoms (we would now say uncountably, infinitely many – Bolzano would probably have been delighted by Cantor's discovery of a hierarchy of infinities). Of these, almost all are aether atoms, and only a finite number are dominant. The dominant atoms in a body are what make matter detectably massive, forceful and resistant, even though they are only finite in number in a finite volume. The boundaries of bodies are not sharp in the sense that there is an abrupt end to the matter. Any dominant atom or collection of them drags a halo or shell (Hülle) of aether atoms around with it, and these can be loosely enough bound to transfer their allegiance from one body to another. For example, when Bolzano picks up his quill to write, some of the aether atoms bound to the quill more than to the surrounding air will be ‘stolen’ by (tend to move with) his fingers. The limits of a body are those aetherial atoms which are the furthest from it that are still attracted to it more than to other, neighbouring dominant atoms. So if the body moves, the aetherial atoms will go with it, unless they are ‘stolen’ by some other more attractive neighbour (PdU 66–7). The boundary of a body will be always changing. What we commonly regard as the touching of bodies is simply their coming into close proximity and their outlying dominant atoms getting too close for comfort (closer than g) and so repelling one another. The intervening space is at all times filled with aether, but that may get compressed in volume.
Bodies made up of different kinds of materials have different natural densities, as is proved by weight measurements of objects of the same volume but different masses, or the same mass but different volumes, it being assumed that weight and mass are always proportional, and weight arises through the mutual attraction of a body and the earth (PdU 63). Mass and density are not simply a measure of the number of atoms in a volume, since all volumes contain the same number of atoms, and a volume of ‘empty space’, that is, containing just aether, has a density far below that of one filled by a body.
As revealed by chemistry, bodies consist of a number of elements or basic stuffs (Grundstoffe). If an atom attracts atoms of kind A more than other kinds, it is said to have an affinity (Wahlverwandtschaft) with the kind A (AP 29). A basic stuff consists (apart from aether) of atoms of a single kind which have an affinity for one another (AP 36). It is affinities that explain chemical reactions (AP 37).
Bolzano's monadology, which we have here summarized in very compressed form, is comprehensive and ingenious, and represents something like the limit to which one can take a monadology in accounting for the dynamical phenomena of classical physics. Bolzano was, through his own reading and his acquaintance with leading scientists of his day such as Doppler, well informed on the state of contemporary science. That subsequent developments in physics and chemistry radically undercut many of the assumptions underlying Bolzano's account is not his fault, though the extent to which he is prepared to treat his account as a-priori metaphysics rather than fallible science is a measure of his questionable rationalism. While characteristic of its time, his assumption of an aether takes on a disturbing tinge of apriorism, in that he regards the lack of direct evidence as insignificant by comparison with non-empirical arguments for the aether. It would have been interesting to know how he would have reacted to the Michelson–Morley null results which indicate the absence of detectable motion of the earth through an aether. Would he have accepted the evidence at face value, or would he have sought an explanation preserving the aether, for example, that the earth drags a large, measurement-neutralizing halo of aether around with it?
In his prefatory notes to AP, Jan Berg summarizes and criticizes Bolzano's metaphysics by noting that it
attempts to explain almost all natural phenomena unitarily and at one go. For this Bolzano needs on the one hand a minimum of very general basic concepts, but on the other hand infinitely many special laws of motion. This dynamic contradiction in Bolzano's metaphysical system seems to imply that no verifiable predictions can be made on its basis.
(BGA 2 A 12,3, 112, my translation)
Since it is by metaphysical design good for everything, it is scientifically good for nothing. This empirical insufficiency, rather than echoes of the official persecution of Bolzano, probably explains adequately enough why there is a clear discrepancy in influence between the mathematical and the physical halves of Paradoxien des Unendlichen. While the physical part is not only now out of date, but had little effect on physics (or metaphysics) in its own day, the mathematical part went on through Dedekind, Cantor, Russell and others materially to inform subsequent developments in continuum theory, set theory and transfinite arithmetic. Why it did not have these effects more directly and rapidly than it easily could have is explained in the Appendix below.2
WORKS BY BOLZANO
AA Athanasia oder Gründe für die Unsterblichkeit der Seele, Sulzbach: J. E. v. Seidel, 1827 [published anonymously]; 2nd improved and enlarged edition (no longer anonymous): Sulzbach: J. E. v. Seidel, 1838; reprint: Frankfurt/M.: Minerva, 1970.
AP Aphorismen zur Physik. In BGA Series II: Nachlaß. A. Nachgelassene Schriften. Vol. 12, 3: Vermischte philosophische und physikalische Schriften 1832–1848. Edited by Jan Berg and Jaromír Loužil. Dritter Teil, 106–148, Stuttgart-Bad Cannstatt: Frommann–Holzboog, 1978.
BGA Bernard-Bolzano-Gesamtausgabe. Stuttgart-Bad Cannstatt: Frommann–Holzboog, 1975 ff.
MWBB the Mathematical Works of Bernard Bolzano. Edited and translated by Steve Russ. Oxford: Oxford University Press, 2004.
PdU Paradoxien des Unendlichen. Edited by Franz Přihonský. Leipzig: Reclam, 1851; reprints: 1889, 1920, 1955, 1964 and 1975. English translations: Paradoxes of the Infinite. Edited by Donald A. Steele. London: Routledge and Kegan Paul, and New Haven: Yale University Press, 1950; and in MWBB, 591–678.
RW Lehrbuch der Religionswissenschaft, ein Abdruck der Vorlesungshefte eines ehemaligen Religionslehrers an einer katholischen Universität, von einigen seiner Schüler gesammelt und herausgegeben, 3 parts in 4 volumes, Sulzbach: J. E. v. Seidel, 1834 [published anonymously]. Critical edition: BGA Series I, Volumes 6–8.
WL Wissenschaftslehre. Versuch einer ausführlichen und grösstentheils neuen Darstellung der Logik mit steter Rücksicht auf deren bisherige Bearbeiter, 4 volumes, Sulzbach: J. E. v. Seidel; 2nd improved edition: Leipzig: Meiner, 1929, 1929, 1930, and 1931; reprints: Aalen: Scientia, 1970 and 1981. Critical edition: BGA Series I, Vols. 11–14. English translation: Theory of Science, translated by Paul Rusnock and Rolf George, 4 volumes, Oxford: Oxford University Press, 2014. References are to the work's sections.
OTHER WORKS CITED
Ganthaler, Heinrich, and Peter Simons. ‘Bernard Bolzanos kosmologischer Gottesbeweis’. Philosophia Naturalis 24 (1987): 469–75.
Morscher, Edgar. Das logische An-sich bei Bernard Bolzano. Salzburg: Pustet, 1973.
Morscher, Edgar. “Robert Zimmermann – der Vermittler von Bolzanos Gedankengut? Zerstörung einer Legende”. In Bolzano und die österreichische Geistesgeschichte, edited by Heinrich Ganthaler and Otto Neumaier, 145–236. St. Augustin: Academia, 1997.
Příhonský, Franz. Atomenlehre des sel. Bolzano. Budissin: E. M. Monse, 1857. Reprinted in Příhonský: Neuer Anti-Kant und Atomenlehre des seligen Bolzano, edited by E. Morscher and C. Thiel. St. Augustin: Academia, 2003.
Winter, Eduard. “Bolzano in Těchobuz-‘Friedenstal’. Ein vormärzliches Idyll unter Polizeiaufsicht”. In Bernard Bolzano 1781–1848. Studien und Quellen. [No editor given], 279–336. Berlin: Akademie-Verlag, 1981.
Zimmermann, Robert. Leibnitz und Herbart. Eine Vergleichung ihrer Monadologien. Eine von der königl. dänischen Gesellschaft der Wissenschaften zu Kopenhagen am 1. Jänner 1848 gekrönte Preisschrift. Vienna: Braumüller, 1849.
Zimmermann, Robert. Philosophische Propadeutik für Obergymnasien. Zweite Abtheilung: Formale Logik für Obergymnasien. Vienna: Braumüller, 1853.
Since Herbart's monadological views are considered elsewhere in this collection, it is worth saying a little about Bolzano's relationship to Herbart and his views. Directly, there is relatively little to report. Herbart being one of the most prominent German philosophers of his day, the conscientious reader Bolzano was well aware of Herbart's views. Herbart is mentioned a number of times in the Wissenschaftslehre, mostly in connection with the theory of ideas and judgements, and mostly critically. Despite ranking Herbart with Fichte, Schelling and Hegel as one of the philosophers he finds make ‘utterances so extraordinary to me that I doubt whether I even understand their correct meaning’ (WL 8), Bolzano nevertheless accords Herbart the epithets ‘acute’ (WL 52) and ‘estimable’ (WL 64), and this, we can trust, without irony. In supporting the idea of his student Fesl that there should be a prize competition to find the best positive criticisms of RW and WL, Bolzano envisaged Herbart as possible chair of the jury, an idea which was undermined by the latter's death in 1841, and later shelved altogether (Winter, ‘Bolzano in Těchobuz-“Friedenstal”’, 294).
Despite some convergences in their views on the physically real, the only serious comment Bolzano makes about Herbart's metaphysics is to dispute, without much attempt at refutatory argument, since clearly he saw it as too obvious, the latter's contention that a simple substance cannot have more than one quality (AP 8).
Herbart did however have, indirectly and unbeknown to either (since both were by then dead), an extremely negative impact on the reception of Bolzano's thought. Bolzano had hoped to find a worthy disciple to carry forward or at least publicize his work in logic and mathematics, and thought to have found one in the young Robert Zimmermann (1824–98), son of a close friend. Bolzano instructed the ‘dear lad’ in mathematics, and when the Austrian school reforms of 1849 under Count Leo Thun and Holenstein (who had studied with Bolzano in Prague) were implemented, Zimmermann wrote a school textbook of logic and psychology that incorporated many of Bolzano's ideas (Zimmermann, Philosophische Propadeutik für Obergymnasien), though without mentioning him as their author (as indeed Bolzano had cautiously counselled). The book was in use only from 1853 to 1860, when it was replaced by a second edition in which Zimmermann largely erased any traces of Bolzano's work. The motivation for this change of mind was that Zimmermann had become a convert to Herbart's views, having published in 1849 a prize-winning study comparing Leibniz's and Herbart's monadologies (Zimmermann, Leibnitz und Herbart), after translating the former's Monadologie into German for the first time (1847). As if this were not enough, Zimmermann voluntarily took on the care of Bolzano's vitally important, ground-breaking and in parts almost print-ready mathematical Nachlass, which he then proceeded to ignore and sit on for the rest of his own (highly successful) career. Far from promoting Bolzano's work, Zimmermann, under the influence of Herbart, rendered Bolzano and posterity a notorious disservice, delaying by many decades the appearance of many of Bolzano's revolutionary ideas (vide Morscher, ‘Robert Zimmermann – der Vermittler von Bolzanos’).
1 In this essay we shall cite Athanasia, ‘Aphorismen zur Physik’ and Paradoxien des Unendlichen using the respective abbreviations AA, AP and PdU. References to AA are to page numbers of the second edition; references to AP and PdU are to Section numbers to enable any edition or translation to be used. Other abbreviations of occasionally cited works are given in the references and are also to sections unless otherwise indicated.
2 I am indebted to Edgar Morscher for help in sourcing primary literature and to two referees for helpful suggestions for improvement.