Abstract
In this paper the way of doing physics typical of mathematical physics is contrasted with the way of doing physics theorised, and practised, by Aristotle, which is not extraneous to mathematics but deals with it in a completely different manner: not as a demonstrative tool but as a reservoir of analogies. These two different uses are the tangible expression of two different underlying metaphysics of mathematics: two incommensurable metaphysics, which give rise to two incommensurable physics. In the first part of this paper this incommensurability is analysed, then Aristotle’s way of using mathematics is clarified in relation to some controversial mathematical passages, and finally the relation between mathematics and the building of theories is discussed.
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Notes
- 1.
In this paper I deal with Aristotle’s account of scientific practice and problem-solving from a purely historical perspective. A comparison with contemporary approaches to the subject would be the much-needed follow-up to this research: I would like to thank Emiliano Ippoliti and Carlo Cellucci for their suggestions in that direction, along with all workshop participants for having discussed the subject with me. Finally, I would like to thank Ramon Masià for having made the diagrams in this article less precarious.
- 2.
Aristotle’s final aim was to extend this principle to all other branches of knowledge, to different degrees, according to the different potentialities of their subject matters. The more a subject deals with necessity and simplicity, the more scientific it can be made (see for instance APr I 30; APo I 27).
- 3.
The same physical situation can be described by more than one model, depending on what kind of quantities one choice to treat as independent variables. A mechanical system of N particles, for instance, admits a classical representation, where the variables are the Cartesian coordinates of the particles involved \(\overline{{r_{i} }} = \left( {x_{i} ,y_{i} ,z_{i} } \right)\) and their time derivatives, or velocities \(\overline{{v_{i} }} = \left( {\frac{{dx_{i} }}{dt},\frac{{dy_{i} }}{dt},\frac{{dz_{i} }}{dt},} \right)\) where i = 1… N. In this case the system is described by Newton’s equations of motion \(\overline{{F_{i} }} = m_{i} \frac{{d^{2} r^{i} }}{{dt^{ 2} }}\), namely N differential equations in 3 variables. But one can use as variables the so-called generalized coordinates \(q_{i}\) and \(\dot{q}_{i} = \frac{{dq_{i} }}{dt}\), where i = 1… 3N, so that the system is described by Lagrangian equations of motion \(\frac{d}{dt}\left( {\frac{\partial L}{{\partial \dot{q}_{i} }}} \right) = \frac{\partial L}{{\partial q_{i} }}\). And one can use also the so-called canonical coordinates \(q_{i}\) and \(p_{i} = \frac{\partial L}{{\partial\dot{q}_{i} }}\), where i = 1… 3N, so that the system is described by Hamiltonian equations of motion \(\frac{\partial H}{{\partial q_{i} }} = - \dot{p}_{i} ; \, \frac{\partial H}{{\partial p_{i} }} = \dot{q}_{i}\).
- 4.
“Quando dunque si facciano simili esperienze in piccole altezze, per sfuggir più, che si può gli accidentari impedimenti de i mezzi, tuttavolta, che noi vediamo, che con l’attenuare, e alleggerire il mezzo, anco nel mezzo dell’aria, che pur è corporeo, e perciò resistente, arriviamo a vedere due mobili sommamente differenti di peso per un breve spazio moversi di velocità niente, o pochissimo differenti, le quali poi siamo certi farsi diverse, non per la gravità, che sempre son l’istesse, ma per gl’impedimenti, e ostacoli del mezzo, che sempre si augmentano, perché non dobbiamo tener per fermo che, rimosso del tutto la gravità, la crassizie, e tutti gli altri impedimenti del mezzo pieno, nel vacuo i metalli tutti, le pietre, i legni, ed insomma tutti i gravi si muovesser coll’istessa velocità?” Galilei (1718), v. III. p. 112.
- 5.
The principle of changing or being changed: Ph. II 1, 192b13-15; III 1, 200b2-13; VIII 3, 253b5-9; VIII 4, 254b16-17; Cael. I 2, 268b16; Metaph. E 1, 1025b18-21.
- 6.
Metaph. E 1, 1025b35-1026a6; K 3, 1061b6 ff.
- 7.
Metaph. M 3, 1077b22-1078a9; Ph. II 2, 193b23-194a12; cf. de An. I 1, 403a15-16; Metaph. N 2, 1090a13-15.
- 8.
Metaph. E 1, 1026a14 ff.
- 9.
Consider for example the “mathematical” treatment of motion in terms of trajectory, velocity and time, which Aristotle develops in depth (see in particular Ph. VI) but which has nothing to do with the essence—that is, the knowledge—of motion in itself, defined in Ph. III in terms of power and act (ἡ τοῦ δυνάμει ὄντος ἐντελέχεια, ᾗ τοιοῦτον, κίνησίς ἐστιν, Ph. III 1, 201a10-11; cf. 201b4-5).
- 10.
When the speeds involved are far from c (the speed of light in void), Einstein’s equations of motion turn into Newton’s laws.
- 11.
Similarities and differences have been studied in Ugaglia (2004), but only the similarities have been taken into account by Rovelli (2015), where any gap is denied and the coincidence between Aristotle’s “equation” of motion and Newton’s limit in a fluid is maintained. Even disregarding any metaphysical obstruction, this conclusion is incompatible with the fact that for Aristotle the speed of a falling body incontrovertibly increases, while in Newton’s approximation it is constant.
- 12.
On the question see Ugaglia (2015).
- 13.
For Aristotle a motion necessarily occurs in a medium. But here we must be careful, because this does not happen accidentally: it is not because Aristotle’s Cosmos does not contain any void that motion occurs in a medium. On the contrary, this is the way in which motion has been defined by Aristotle, namely as a relative notion, which requires the presence of a medium. In other words, for Aristotle the absence of void is not an imposition, but a logical consequence of his physical premises.
- 14.
The closer the movable object is to the end of its motion, and the more of the form it was lacking—and tends to—has been acquired, the more efficient its motion is (Ph. VIII 9, 265b12-16. Cf. Ph. V 5, 230b25-26; Cael. I 8, 277a27-29; II 5, 288a20-21).
- 15.
In a formula \(v\left( t \right) = \sqrt {\frac{1}{C}} \sqrt {\frac{{W_{b} }}{{\rho_{m} }}}\) where C is a constant coefficient, depending on the shape of the moving object, W b is the absolute weight of the moving object and \(\rho_m\) is the density of the medium.
- 16.
Perhaps, the fact that two synonymous terms—physics and natural philosophy—have been traditionally employed to denote the same object has contributed to create this misunderstanding among non-specialists.
- 17.
Take, once again, Aristotle’s treatment of local motion, which is one of the most “formalised” branches of his physics: the qualitative, nonformalised theory of motion occupies the whole book IV of the De Caelo, together with long sections of book III and II of the Physics, while the quantitative features of motion are briefly mentioned in individual passages, all belonging—and designed to underpin—arguments other than those concerning motion. In order to obtain the “true” physics, in this case we must not only collect these scattered hints, but translate them into formulae, which is anything but a safe operation, as the simple example of speed in Sect. 2.3 shows.
- 18.
The operation is called ἀφαίρεσις, this being the standard term for subtraction in Greek mathematics. It must be noted that what is subtracted is not the mathematical object, but the more “physical” features of the physical object, namely the ones connected to change. Indeed, in order to subtract something, one has to know the thing to be subtracted, but to know mathematical objects before knowing physical ones leads to a Platonic position, very far from Aristotle’s one. For this reason I prefer to avoid the term abstraction, too much connoted in this sense.
- 19.
See in particular APo I 2. In addition, Aristotle says that mathematics speaks with the greatest accuracy and simplicity (Metaph. Μ 3, 1078a9-11), and it speaks about the beautiful and the good (Metaph. Μ 3, 1078a32-b6). See Metaph. M and N for more general statements about mathematics contra Plato.
- 20.
τὴν δ′ ἀκριβολογίαν τὴν μαθηματικὴν οὐκ ἐν ἅπασιν ἀπαιτητέον, ἀλλ’ ἐν τοῖς μὴ ἔχουσιν ὕλην (Metaph. α 3, 995a14-15); cf. APo I 8, passim.
- 21.
- 22.
Aristotle’s theory of motion strikes interpreters as being amazingly naive. It certainly is, but only if it is read from a modern “kinematic” perspective, that is, against what has been the standard notional and argumentative background at least from Philoponus onwards. If one adopts the seemingly common-sense perspective that motion is essentially a translation in a space, possibly but not necessarily filled with matter, misunderstandings are unavoidable. Crucial notions such as lightness, for instance, or the non-existence of vacuum, must be introduced as ad hoc hypotheses. On the contrary, no ad hoc hypotheses are needed if one sets out from a model of Aristotle’s theory of motion within the framework of hydrostatics: imagine that Aristotle devised his theory of motion in water. Less empirically, imagine that he envisaged it by analogy with what happens in water, that is, by analogy with hydrostatics, understood as a branch of mathematics. Granted, hydrostatics studies systems in a state of equilibrium, but through some extrapolation the processes of reaching equilibrium might lead to some interesting suggestions about how natural bodies move toward their proper places. The hydrostatical origin of Aristotle’s theory of motion is discussed in Ugaglia (2004, 2015).
- 23.
It is crucial to bear this distinction in mind, in order to avoid a common misunderstanding: the partial and erroneous view of ancient geometric analysis constituting the scholarly vulgata, for example, is in some measure due to a confusion between the mathematician’s practice and formalisation; in turn, this confusion is due to an improper reading of Aristotle’s mathematical passages (see, for example, the current interpretation of EN III 3).
- 24.
See APr I 41, 50a1-2.
- 25.
“We must also distinguish certain senses of potentiality and actuality; for so far we have been using these terms quite generally. One sense of “knower” is that in which we might call a human being knower because he is one of a class of educated persons who have knowledge; but there is another sense, in which we call knower a person who knows (say) grammar. Each of these two persons has a capacity for knowledge, but in a different sense: the former, because the class to which he belongs is of a certain kind, the latter, because he is capable of exercising his knowledge whenever he likes, provided that external causes do not prevent him. But there is a third kind of educated person: the human being who is already exercising his knowledge: he actually knows and understands this letter A in the strict sense. The first two human beings are both knowers only potentially, whereas the third one becomes so in actuality through a qualitative alteration by means of learning, and after many changes from contrary states ‹of learning›, he passes from the inactive possession of arithmetic or grammar to the exercising of it.” (de An. II 5, 417a21-417b2, the translation is Hett’s, modified).
- 26.
Metaph. Θ 1, 1046a9-13; cf. Δ 12, 1019a15-21 and 1019b35-1020a6; Θ 6, 1048a28-29; Θ 8, 1049b5-10.
- 27.
Metaph. Θ 8, passim.
- 28.
Henceforth, the indices are omitted when the argument holds indifferently in case 1 and 2.
- 29.
In this case, a diagram must be conceived not as a result—that is, as part (the right κατασκευή) of a formalized proof—but as a work in progress. Aristotle is not interested in the final diagram—such as those accompanying proofs in Euclid’s Elements—but in the construction viewed in its process of development.
- 30.
Metaph. Θ 9, 1051a21-33.
- 31.
APo II 11, 94a28-34.
- 32.
Top. VIII 1, 157a1-3: ἔτι τὸ μηκύνειν καὶ παρεμβάλλειν τὰ μηδὲν χρήσιμα πρὸς τὸν λόγον, καθάπερ οἱ ψευδογραφοῦντες. πολλῶν γὰρ ὄντων ἄδηλον ἐν ὁποίῳ τὸ ψεῦδος «next, stretch out your argument and throw in things of no use towards it, as those who draw fake diagrams do (for when there are many details, it is not clear in which the error lies)» The translation is Smith’s.
- 33.
Here it is not a question of seeing something (a triangle) and not knowing that this something is a triangle, but rather of seeing nothing (interesting) at all. The geometer, sitting in front of his figure, would like to see a triangle, or some other figure, to which he could apply some (known) results. But he does not see anything interesting.
- 34.
APo II passim.
- 35.
In this sense, a simple answer can be given to interpreters who act surprised when they find in Aristotle’s Analytics not a prescription for the construction of demonstrations or for the acquisition of knowledge, but a definition of what a demonstration, or knowledge, is.
- 36.
If I do not know what I am searching for, I cannot correctly articulate my question, and my search is very difficult, as Aristotle explains in Metaph. Ζ 17 (cfr. Β 1).
- 37.
I adopt here the interpretation of nous and epistēmē as intellectual states, as maintained for instance in Barnes (1993).
- 38.
See note 22 before.
- 39.
It is important to point out that for Aristotle the elements of a science are terms, and not propositions.
- 40.
APr I 23 passim.
- 41.
Ἡ μὲν οὖν ὁδὸς κατὰ πάντων ἡ αὐτὴ καὶ περὶ φιλοσοφίαν καὶ περὶ τέχνην ὁποιανοῦν καὶ μάθημα (APr I 30, 46a 3-4). On the fact that in PS not only formal connections, but also material connections or arguments grounded on signs and common opinions are acceptable, see for example APo I 6; APr II 23-27; Top. I 10 and I 14; HA passim. Consider in particular Historia animalium: with the big collection of material contained here, which consists of purely material connections, it is impossible to build a theoretical system. In order to convey this material in a scientific system it is necessary to give a “direction” to the results obtained, introducing a hypothetical necessity which steers the process of deduction.
- 42.
- 43.
APo I 33-34; cfr. EN VI 5.
- 44.
APr I 27-31; Top. I 2; APo II 11-17; SE 16; EN VI 7.
- 45.
In fact, this may be a controversial point. At least some of the procedures Aristotle describes are the object of interesting studies in the contemporary philosophy of science; see for instance Magnani (2001) on the notion of abduction. More generally, it might be interesting to contrast Aristotle’s conception of problem solving, and its relation to logic, with the recent developments in the studies of theory-building as problem-solving, where a logical toolbox for such a kind of procedure is proposed and discussed. See in particular Cellucci (2013), Ippoliti (2014) and the papers contained in this volume.
- 46.
In particular, In Prior Analytics Aristotle explains what a syllogism is (I 1-26 and II 1-22); how it must be constructed, i.e. how two given terms can be related (I 27-31 and II 23-27), and how a syllogism can be reduced to another, i.e. how the relation between the two terms can be usefully changed into another relation, in order to obtain a more scientific result (I 32-45).
- 47.
APo I 9.
- 48.
In this perspective one can say that the actual subject of Prior Analytics is problem-solving, while that of Posterior Analytics is scientific problem-solving. If all the sciences were like mathematics, there would be no need for Prior Analytics: since every mathematical proof may be reduced to a Barbara syllogism, it is sufficient to prove the relation in a general case and it will hold in any other (Top. II 3, 110a29), and will be easy to convert. On scientific problem-solving in Aristotle see Mendell (1998). The most complete account of the problem-solving approach in ancient Greek mathematics is Knorr (1986). On its relation to analysis, see Mäenpää (1993).
- 49.
Notice that, to reduce physical demonstrations to mathematical proofs would be to introduce determinism in nature, a hypothesis which Aristotle firmly rejects. On Aristotle’s teleology as an alternative to determinism, see Quarantotto (2005).
- 50.
For Aristotle mathematical proofs can be reduced to syllogisms in Barbara ( APo I 14). On the subject see Mendell (1998).
- 51.
- 52.
“Thus, by calculating that signification according to the algebraic method used here, namely: \(\frac{{S\left( {\text{signifier}} \right)}}{{s\left( {\text{signified}} \right)}} = s\left( {{\text{the}}\,{\text{statement}}} \right)\) with \(S = - 1\) produces \(s = \sqrt { - 1}\). […] Thus the erectile organ comes to symbolize the place of jouissance, not in itself, or even in the form of an image, but as a part lacking in the desired image: that is why it is equivalent to the \(\sqrt { - 1}\) of the signification produced above, of the jouissance that it restores by the coefficient of its statement to the function of lack of signifier (−1)” (Lacan 1977). On this and other cases of misuse of mathematics see Sokal and Bricmont (1998).
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Ugaglia, M. (2018). “Take the Case of a Geometer…” Mathematical Analogies and Building Theories in Aristotle. In: Danks, D., Ippoliti, E. (eds) Building Theories. Studies in Applied Philosophy, Epistemology and Rational Ethics, vol 41. Springer, Cham. https://doi.org/10.1007/978-3-319-72787-5_7
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