¶ 1 Leave a comment on Absatz 1 0 If experience is regarded as a requirement for knowledge, a consistent explanation of mathematical knowledge poses special problems. This does not primarily concern the question whether mathematical truths can be known a priori, but rather their peculiar justificational status. Early modern Aristotelians acknowledge that mathematical truths are self-evident: They can be understood ‚at a glance‘. And mathematical objects are abstractions: we do not need (and maybe even cannot have) an experience of ‚a length without breadth or depth‘. Nevertheless, we are ready to accept this as the definition of a mathematical line without hesitation.
¶ 2 Leave a comment on Absatz 2 0 Javellus discusses the self-evidence of mathematical truths in the context of an objection that states the incompatibility of Javellus’s thesis that experience is always a necessary instrument for knowledge with the special status of mathematical truths. This is a difficult question, because knowledge of mathematical objects may be based on a single perception, while experience seems to require multiple perceptions of the same (nexus of) object(s).
¶ 3 Leave a comment on Absatz 3 0 […] si quando ars, et scientia acquiruntur per inventionem, necesse est ut semper acquirantur per experimentum, et videtur quod non, nam si quis semel tum viderit, quomodo triangulus habet tres aequales duobus rectis, vel quomodo contingit super lineam datam, triangulum constituere aequilaterum, per certitudinem cognoscet semper sic esse, ex consequenti de hac conclusione triangulus habet tres etc. habebit scientiam. et non per experimentum, cum non fit experimentum, per unam sensationem, vel per unam memoriam tantum, sed per multas, ut dicit philosophus in textu, […] (Javellus, fol. 17 r
¶ 4 Leave a comment on Absatz 4 0 Under the premise that art and science are acquired through invention, it is necessary that they are always acquired by experience. It seems that this is not the case: if someone saw only once, in which way the three angles of a triangle are equal to two right angles, or how to construe an equilateral triangle on a given line, he would cognise with certainty that this is always the case. Therefore, he would have knowledge about the conclusion ‚a triangle has three equal …‘. This knowledge does not come about through experience, because it [sc. experience] does not come about by one sensation or only one representation in memory, but by many, as the philosopher [sc. Aristotle] says in the text.
¶ 5 Leave a comment on Absatz 5 0 Javellus replies that we do require experience for mathematical knowledge, because the objection does not take into account that there are two forms of experience: ‚virtual‘ and ‚formal‘ experience. Formal experience (not the experience of forms, but experience in the strict sense of the word) consists in multiple perceptions. Virtual experience is a state that requires only one perception, but its efficacy (virtus) equals that of formal experience: Whether I perceive an immutable object only once or whether I perceive it several times does not change my apprehension of this object. Perceptions of mathematical objects belong to this class.
¶ 6 Leave a comment on Absatz 6 0 Respondeo, dico quod semper requiritur experimentum. Sed adverte, quod est duplex, sive formale, et virtuale, formale est quod fit ex pluribus sensationibus eiusdem rei, et multis memoriis eiusdem. virtuale autem est, quod fit unica sensatione, et unica memoria, quae licet in se sit una tantum, aequivalet tamen multis, nam sunt quaedam res adeo stabiles, quod semel vidisse quomodo se habent, est semper vidisse, Quaedam autem sunt adeo variabiles, quod nisi multiplicatur earum sensationes, respectu eiusdem effectus, parva aut nulla potest haberi de eius certitudo. et tales sunt rerum inferiorum naturae. In primis sufficit experimentum virtuale. In secundis autem requiritur formale, et cum in gradu primo reponuntur mathematica, ideo in eis sufficit experimentum virtuale. In naturalibus autem requiritur formale. (Javellus, fol. 17 r sq.)
¶ 7 Leave a comment on Absatz 7 0 I reply that experience is always required. But the reader may note that it [sc. the concept of experience] is ambiguous, because it means either formal or virtual experience. Formal experience results from several perceptions of the same thing or many representations of it in memory. Virtual experience results from only one perception or one representation in memory. But even though it is in itself only one [sc. representation], it is equivalent to many [sc. representations]. Some things are to such an extent unchangeable that having seen once how they behave is [sc. the same] as having always seen them, others are quite variable so that unless perceptions of them regarding the effects of the same thing are multiplied, we have only marginal certainty regarding them or no certainty at all. Such are the essences of inferior [sc. sublunar] things. For the first [sc. kind of objects] virtual experience is sufficient. Regarding the second [sc. kind of objects] formal experience is required. And since mathematical objects belong to the first class, for them virtual experience is sufficient. For natural things formal experience is required.
¶ 8 Leave a comment on Absatz 8 0 So Javellus sees a significant difference between objects of natural philosophy and objects of mathematics: The immutability of mathematical objects allows for a different mode of experiencing them than the mutability of sublunar nature. But nevertheless, all knowledge concerning mathematical or natural objects is based on experience.
¶ 9 Leave a comment on Absatz 9 0 The objection discussed by Fonseca points into a slightly different direction: Here, geometrical definitions are introduced as being self-evident. Since they must be regarded as principles, there are principles that do not require experiential confirmation (contrary to what Fonseca himself asserts).
¶ 10 Leave a comment on Absatz 10 0 Praeterea, definitiones geometricae, quae ante theoremata ponuntur, ut quid sit linea, quid triangulus, sunt quaedam principia: haec autem non indigent experimento, cum iis propositis nemo fit, qui non statim assentiatur: igitur non omnia principia proprio indigent experimento.(Fonseca, col. 97)
¶ 11 Leave a comment on Absatz 11 0 Besides that, geometrical definitions, which are proposed before any theorems, e. g. what is a line, what is a triangle, are in some sense principles. They do not, however, require experience, because if they are articulated, everyone will assent to them instantaneously. Therefore, not all principles require properly experience.
¶ 12 Leave a comment on Absatz 12 0 Fonseca denies that it makes sense to regard mathematical definitions as principles in the strict sense of the word. They clarify the meaning of mathematical terms (what Javellus would call the quid nominis), not a purported quid rei. So they are just a didactic ploy: It would be feasible, though impractical, to insert these explanations into the process of mathematical deduction itself. This would not stand in the way of the conclusiveness of such a deduction.
¶ 13 Leave a comment on Absatz 13 0 Ad tertium, quod attinet ad principia propria, dicendum est, conclusionem nostram non esse intelligendam de iis propriis principiis, quae sunt merae positiones, hoc est, quae solum permittuntur ad explicanda nomina, quibus in demonstrationibus utendum est: quales sunt definitiones, quae initio Geometriae ponuntur. nec enim praemittuntur ad declarandas rerum essentias, sed ut is, cui fit demonstratio, intelliget, quid Geometra significare velit vocabulo lineae, trianguli, et caeteris: ne demonstrator in ipso demonstrationis cursu cogatur ea explicare. (Fonseca, col. 98)
¶ 14 Leave a comment on Absatz 14 0 Regarding the third [sc. objection] that pertains to principles of particular sciences it must be said that our conclusion should not be understood as stating that it includes those particular principles that are just constructions. That means that they only allow the explication of names which are used in demonstrations, such as the definitions standing at the beginning of geometry. They are not mentioned in advance in order to explain the essences of things. Rather they are used so that the recipient of a demonstration may understand what the geometrician wants to express by using the words ‚line‘, ‚triangle‘ etc. and the person conducting the proof need not explain it in the process of proving itself.
¶ 15 Leave a comment on Absatz 15 0 Since mathematical definitions only clarify the use of mathematical language and have merely an expository function, they must be regarded as linguistic conventions. Since linguistic conventions seem to be arbitrary, they do not tell us anything about how the world is. Experience does. Hence mathematical principles cannot be based on experience – and they need not be, because they have no epistemic value beyond the confines of mathematics proper.
¶ 16 Leave a comment on Absatz 16 0 Quo fit, ut nihilominus Geometra demonstret sua theoremata, etsi definitiones, quas posuit, non explicent essentias rerum, modo per eas constet, quid vocabulis, quibus inter ipsos disseritur, significetur. Hac autem de causa appellaverim huiusmodi definitiones, meras positiones, quia sunt velut conventiones quaedam inter eos, qui disserunt, quasi dicant, ponamus nomine lineae significari longitudinem sine latitudine, et profunditate; nomine trianguli figuram planam tribus lineis contentam: et ita in caeteris. Cum ergo principia conventione sola constent, satis perspicuum est, nullo opus esse experimento, ut illis assentiamur. (Fonseca, col. 98)
¶ 17 Leave a comment on Absatz 17 0 Nevertheless the geometrician does prove his theorems, even though the definitions which he assumes do not explain the essences of things. He only settles by them what the words mean they [sc. the geometricians] use among themselves. For this reason, I would call these definitions mere constructions, because they are like conventions of those who talk [sc. about geometry] and they just mean ‚let’s assume that the word ‚line‘ signifies a length without breadth or depth, the word ‚triangle‘ a plane figure contained by three lines, etc.‘. Because these principles are purely conventional, it is easily seen that we need no experience in order to assent to them.
¶ 18 Leave a comment on Absatz 18 0 So Fonseca is a conventionalist about mathematics: Definitions of mathematical terms are mere linguistic clarifications of their use. They have no ontological import. Mathematical objects are constructions. Constructions cannot and need not be experienced in order to be used in proofs.
¶ 19 Leave a comment on Absatz 19 0 According to Javellus, mathematics is no exception from the rule that experience is required for knowledge, because there is a form of experience that can take into account the self-evidence of mathematical knowledge: A one-time-perception of a mathematical object is epistemically equivalent to multiple perceptions of a changeable object – one-time-perceptions of mathematical objects are ‚virtual experiences‘.