(Note from Dave: Tex embedded a lot of images in his document and used footnotes that I was not able to transfer to the blog...so you will have to use your imagination.)
Nature, at first glance, seems like it can be simply understood. The minutiae might take time, but reaching a general law about something like the rate at which bodies fall cannot be very difficult. It does not take much reading of Galileo’s Two New Sciences to find out that this is not the case. The reader quickly understands that in order to arrive at any laws, a method must be put forth. If the method is good, then it should allow us to arrive at the correct answer consistently. But the method must also be comprehensible. Nature is studied by humans, so any method of studying it must not only be grounded in reason, but take into account ineffable qualities, such as prejudice and confidence.
Nature, at first glance, seems like it can be simply understood. The minutiae might take time, but reaching a general law about something like the rate at which bodies fall cannot be very difficult. It does not take much reading of Galileo’s Two New Sciences to find out that this is not the case. The reader quickly understands that in order to arrive at any laws, a method must be put forth. If the method is good, then it should allow us to arrive at the correct answer consistently. But the method must also be comprehensible. Nature is studied by humans, so any method of studying it must not only be grounded in reason, but take into account ineffable qualities, such as prejudice and confidence.
Of the three characters in Two New Sciences, Simplicio has the most difficulty comprehending the method presented. With each new problem, he seeks the answer and inevitably returns empty-handed. This essay looks at the mistakes of Sagredo and Simplicio, who often have trouble answering the problems put forth by Salviati. Through this exploration, we want to determine exactly how comprehensible the Galilean approach to nature is, and the penalties of having a method that may only be understood by a few people. Sagredo's “Fancy” and its Flaws
Immediately after Salviati finishes reading the second proof in On Naturally Accelerated Motion (Proposition 2.II)[1],[2] and its corollary, Sagredo interrupts Salviati's reading to develop a “Fancy.” 2.II[3] is the first time we see a presentation of the law of uniform acceleration. It states that a falling body increases in distance according to the square of the time of the fall. It is hard to call Sagredo's Fancy a proof in the classic sense, but Sagredo and Simplicio think it finds the law of uniform acceleration equally well. And Simplicio, at least, prefers the Fancy. He tells the other interlocutors after Sagredo presents the Fancy, Really I have taken more pleasure from this simple and clear reasoning of Sagredo's than from the (for me) more obscure demonstration of the Author, so that I am better able to see why the matter must proceed in this way, once the definition of uniformly accelerated motion has been postulated and accepted.[4]
Is Simplicio allowed to view the Fancy and the Author's proof as interchangeable, or is the Fancy less conclusive? Sagredo begins the Fancy by asking us to take a look at an image[5]. We must imagine the section below line FI as not pictured at first. Given time AI, and an arbitrary angle IAF, Sagredo cuts the line AI at C, also cutting AF at B. Sagredo assumes “without argument,” that Proposition 2.I[6] holds. Proposition 2.I effectively says, in the context of the Fancy, that over the given time AC a body moving in uniform motion with half of the final speed BC will cover the same amount of space. This space traversed in uniform motion is represented in the Fancy by the rectangle DC. Proposition 2.I is the link which binds the first treatise, which treats uniform motion, with the two later treatises, which treat of accelerated motion. 2.I is the link which binds all three treatises together. At this point, Sagredo has redrawn Proposition 2.I, and assumed its enunciation. Now he tries to prove the law of uniform acceleration from this picture. The Fancy can be summarized thus: If a uniformly accelerating body continues to accelerate uniformly (so if the hypotenuse of the triangle continues to be extended in a straight line), the horizontal IF, which represents the speed, will necessarily increase. And it happens that the rectangular spaces created along IF will increase in a series of odd numbers. At time C, there is one area ADCE. At time I, which is exactly double the time at C, the area CBNI is triple that of the previous area. After the next equal time interval IO, this area has expanded to five times the original (area IFRO). If we keep adding, we will continue producing an increase in area by odd numbers–7,9,11,etc. If we then add these odd numbers, the sums are perfect squares (i.e. 1+3=4, 1+3+5=9, 1+3+5+7=16, etc.). In the same way, if we add up the rectangles at equal time intervals in Sagredo's picture, the total number of rectangles which make up the triangle always equals a perfect square corresponding to the amount of equal vertical spaces covered (i.e. AC2=12, CI2=22, IO2=32, etc.)
Sagredo has come to the same conclusion as the Author came to in Corollary 1 to 2.II. Corollary 1 reads, “Thus when the degrees of speed are increased in equal times according to the series of natural numbers, the spaces run through in the same times undergo increases according with the series of odd numbers from unity.”[7]
Why are two separate methods used to teach the law of uniform acceleration? Should one method be considered “better,” and how would we judge which method is better? The method exhibited by Simplicio and Salviati is more intuitive. The formal method is more rigorous. In the formal method, the proofs are formulated by Galileo and bear a genealogy which stretches back to Euclid. Because Galileo's method is the orthodox one, it carries the weight of certainty. Should Salviati have told Simplicio that, despite taking pleasure in the Fancy, he should continue to focus his studies on the Treatise?
Sagredo's Fancy was likely intended by Galileo to be an alternate approach for less-educated readers to understand the law of uniform acceleration from Proposition 2.I. Proposition 2.II is the formal solution to this challenge. The Fancy, on the other hand, is the informal solution, and the one preferred by Sagredo and Simplicio, whose lack of formal mathematical training is well established at this point in the dialogue. Given that we are seeking a physical law, it is necessary that the proof of the law represent a physical space. Sagredo apparently believes the Fancy can work as a physical proof, but the connection is lacking.
Since the hypotenuse AP in the Fancy must also represent the uniform acceleration, it would be nonsensical for the hypotenuse of a triangle not to be continued in a straight line, for the object created would no longer be a triangle. In the Fancy, the representation of uniform acceleration as the continuously straight hypotenuse of a triangle is drawn directly from the image given in Proposition 2.I. If the vertical element of the triangle continues to extend in equal increments, and continues producing a replication of Proposition 2.I along every equal segment of the hypotenuse–as Sagredo does with triangle BFN and triangle PFQ–you will only arrive at a conclusion about the space contained within a triangle, not any physical law.
There is another difference between the Author's proof and Sagredo's Fancy – in the Fancy there is no decent representation of the space traversed. Both proofs represent time along the vertical axis and speed along the horizontal, but the Fancy lacks a separate representation of the space covered. Sagredo considers his conclusions about the expanding areas ADEC, BNCI, FRIO to be equivalent, but it is impossible to model Sagredo's Fancy in a physical situation, such as the inclined plane experiment.
Without an adequate physical representation, our certainty about the law comes from the mathematical object and not the physical situation. This need for a physical representation is not for convenience. The expansion of areas in the Fancy is a property we could find by dividing up any triangle, whereas the physical representation Sagredo interprets into the Fancy is only nominal.
The Author's proof is better because it represents the space being covered with a separate linear magnitude, bringing us one step closer to the physical certainty we seek. Also, its proof rests on something proved in On Equable Motion,
Now in Proposition IV of Book I it was demonstrated that the spaces run through by moveables carried in equable motion have to one another the ratio compounded from the ratio of speeds and from the ratio of times, since the ratio of one-half PE to one-half OD, or of PE to OD, is that of AE to AD. Hence the ratio of spaces run through is the duplicate ratio of the times; which was to be demonstrated.[8]
In other words, if we want to figure out the space two bodies moving in equable motion with different speeds and different times will cover, we need to compound the ratio of the two speeds along with the ratio of the two times. The proof is written with the understanding that the reader has probably looked at Book Five of Elements.
In Proposition 2.II, Galileo relates unequal lengths HM and HL, which represent spaces, to Proposition 1.IV, which tells us how to find the ratio of two spaces traversed in equable motion at unequal speeds. The connection made in Proposition 2.I between uniform and accelerated motion means that all the proofs in On Naturally Accelerated Motion can be proved from the definitions, axioms, and propositions in On Equable Motion.
Proposition 2.II has a connection to the proofs that came before it. On Uniform Motion begins like Euclid's Elements, with a definition and set of axioms. It then moves to simple, evident propositions, and finally to more complicated ones. Thus, Proposition 1.IV, which is more complicated and less intuitive than Proposition 1.I, is built from simple, axiomatic statements. Even when we reach Proposition 2.II, it is set up in such a way that its conclusion follows from previously understood truths. This is similar to our journey to Elements I.47: though I.47 is not immediately obvious, we build to it from previously accepted principles.
If presented with only Proposition 2.I and Sagredo's Fancy, I don't think we could be intellectually certain of the law of uniform acceleration in the same way we are after Proposition 2.II. Neither Proposition 2.II nor Sagredo's Fancy gives us physical certainty about the law–we need the inclined plane experiment for that. The Fancy does not give an explanation of what we might observe in the inclined plane experiment in the same way I think 2.II does, since the conclusion of the Fancy necessarily falls out from the image that has been constructed through Proposition 2.I. In the author's proof, the image works as an aid to understanding.
The extra difficulty in Two New Sciences, or any physical treatise, is that everything proved cannot just be proved intellectually; it must also be proved empirically. In physics, a mathematically consistent proof has no relevance without an experimental application. The division between mathematical certainty and physical application is easily bridged in Galileo's simpler proofs, such as Proposition 1.III,[9] where a simple experiment will show us that the proposition is empirically true. If all the propositions leading up to Proposition 2.II are true, and if Galileo's axioms produce a consistent formal system, then Proposition 2.II should be true, as long as physical bodies behave according to a set of predictable laws, as opposed to behaving randomly.
Throughout On Equable Motion, Galileo connects the physical and the intellectual by treating basic, observable physical situations as mathematical axioms. Using the premise that nature behaves as a consistent system, true conclusions are reached which are not immediately apparent. As long as we stay within the system laid out by Galileo, we shouldn't fear its results.
Sagredo's Fancy is outside of Author's system (yet in his book). It does have Proposition 2.I in mind, but it is not built within any apparent axiomatic system, let alone Author's. In the Author's treatises, any new truth is deduced from something already known, in a true Cartesian manner. The image only assists us. For Sagredo, the image is the proof.
The Fancy does not rely on complicated (for Simplicio, at least) mathematical tools such as compound ratio. But for this reason, it is hard to see how Sagredo's Fancy could be used in the later proofs of On Naturally Accelerated Motion, since it is not built upon axioms. Simplicio sees the Fancy as “simple and clear,” while Galileo's proof is “obscure.” The Author relies on obscure mathematical tools, while the Fancy presents everything needed to understand it in the image. The Fancy makes things more clear for people like Simplicio, but it also restricts the creation of new knowledge.
These thoughts impel me to read Sagredo's Fancy as a warning. Galileo himself probably does not view Sagredo's conclusion as anything more than a curiosity, and expects that an educated reader will feel the same way. Salviati has no response to the Fancy, but if we were staging this dialogue, what expression would he have on his face? Would it be one of disgust or annoyance? Or would he show the earnest praise of a teacher marveling at a student's cleverness?
This section began by quoting the first part of Simplicio's response to the Fancy, and it will end with the second half of the response,
But I am still doubtful whether this is the acceleration employed by nature in the motion of her falling heavy bodies. Hence, for my understanding and for that of other people like me, I think that it would be suitable at this place [for you] to adduce some experiment from those (of which you have said that there are many) that agree in various cases with the demonstrated conclusions.[10]
|
 |
| ||||
| ||||
Simplicio says that he “takes more pleasure” in Sagredo's Fancy, but whatever pleasure he derives from the proof should be sacrificed at the altar of experiment. Simplicio treats method as a matter of taste and does not appear concerned that one method might better account for what we experience. Salviati does not respond to the Fancy, but maybe the look on his face is smug; he knows his method is correct. After reading Sagredo's Fancy, we are left asking: Is there a correct or better method for studying nature?
When the interlocutors move into Day Four, to study On the Motion of Projectiles, we will see that Sagredo and Simplicio are approaching, or have reached, a limit of understanding. Their method ceases to yield fruit. An axiomatic system such as Euclid's Elements can be applied by Galileo 2,000 years later, yet it wouldn't be useful to teach Sagredo's Fancy to school-children.
Projectile Motion: Technical and Philosophical Hurdles
For Simplicio, the pleasure and simplicity of the Fancy find their opposite in the first proposition of On the Motion of Projectiles (Proposition 3.I), at the beginning of Day Four. The proposition reads as follows: “When a projectile is carried in motion compounded from 
equable horizontal and from naturally accelerated downward [motions], it describes a semiparabolic line in its movement.”[11] The proposition is a synthesis of the concepts of uniform and accelerated motion presented in the first two treatises, but Simplicio's difficulty comes in the final clause of the proof; Simplicio, along with Sagredo, is not versed in Apollonius. The dialogue between the enunciation and the proof presents Salviati's efforts to give the other two the knowledge required to proceed. The attempt appears beneficial to Sagredo, but Simplicio has much more trouble, and is genuinely at a loss. The basic terms of Apollonian geometry are new to him. He says, “And unless that little that I have learned of geometry from Euclid, after our other discussions a long time ago, will be sufficient to render me capable of what is required for understanding the demonstrations to come, I shall have to content myself with merely believing the propositions without comprehending them.[12]”
We can question whether Simplicio comprehended previous propositions, given his consideration of 2.II as obscure, but he has now reached a limit of understanding which he only approached at earlier points in the dialogue. He can no longer understand the proofs, but only believe them. Simplicio speaks a little more after Proposition 3.I is read by Salviati, but he never makes it clear that he fully understands the Author's proof.
While Simplicio is struggling, it also becomes more clear that the intuitive method he favored in Sagredo's Fancy is losing relevance. For Proposition 3.I, there is no apparent equivalent to Sagredo's Fancy, where an intuitive approach that may appeal to an untrained student like Simplicio can be presented alongside the formal proof. The dialogue between the reading of the enunciation and the reading of the proof serves as a warning that specific knowledge is needed to grasp both the lemmas and the proof itself. Salviati seems confident that by presenting the lemmas Simplicio will be satisfied. When Simplicio does not understand, there is no recourse–basic knowledge of conic sections is required for the barest understanding of this proof.
For Proposition 3.I to exist at all, it requires Galileo's method, which is based in mathematical demonstration. Insofar as it can be called a method, the intuitive approach favored by Sagredo and Simplicio has lost any application, while Galileo's method continues to produce new findings. The good scientist who wants to unearth new truths in nature must follow a method that is not only formalized, but one that will make use of specialized mathematical knowledge (the example in this case being the two lemmas from Apollonius). Intuitive, informal reasonings like Sagredo's Fancy have lost their usefulness by Day Four.
Despite the great hope the Galilean method contains for the scientist, a penalty comes along with its application–the knowledge is not accessible to all. Simplicio is reduced to “believing without comprehending.” It is true that Simplicio could teach himself Euclid and Apollonius, but there is a limit to how much he can learn these old things while still learning the new sciences that are being developed. With the creation of more new knowledge on top of old, the benchmark of understanding will continue to be raised.
Simplicio's prior education does him no favors in learning these new sciences. After the parabola is introduced, he says, “For although our philosophers have treated this matter of projectile motion, I don't recall that they felt themselves obliged to define the lines described thereby, other than in very general terms–that these are curved lines, except for things thrown vertically upward.”[13] He acknowledges a similar issue in Day One, after being taught that the surface area and volume of a solid do not increase or diminish in a 1:1 ratio. “[Y]ou may both believe me that if I were to begin my studies over again, I should try to follow the advice of Plato and commence from mathematics, which proceeds so carefully, and does not admit as certain anything except what it has conclusively proved.”[14]
The quote above implies that something more obstinate than time is holding Simplicio down. Given an infinite amount of time, Simplicio could teach himself enough mathematics to comprehend accurately the Author's proofs. He desires a better understanding of the material, yet he always reverts back to his education as an Aristotelian, a background which leaves him unprepared to think about nature in a Galilean manner.
Simplicio shows a desire to understand, and Salviati is patient with him, but tension remains. These new sciences are built on a simple premise: the certainty of geometry can be applied to a study of nature, which in turn makes the study of nature more certain. Natural phenomena can be proved.[15] This approach allows us to know things which were previously uncertain, such as the shape in which any projectile will travel. At the same time, this method alienates those who are of a certain mental caste. Salviati is not Socrates–he cannot bring Simplicio to greater knowledge simply by questioning. Instead, the knowledge must be presented.
Despite Simplicio's timidity, Salviati goes on to read Proposition 3.I. It is immediately after this reading, though, that Sagredo and Simplicio mount one of the most robust attacks in all of Two New Sciences. Just when Simplicio appeared to be completely defeated, he and Sagredo expose a crucial fissure in the Galilean method. A principle of the Galilean method is its geometrical nature; physical proofs are given the certainty of geometry. Prior to Day Four, there was little question of whether this fusion of abstract and concrete was always valid, since it had worked so well.
Immediately after Proposition 3.I, Sagredo and Simplicio question the validity of this method as applied to projectile motion, and Salviati struggles to respond. Between them, they offer three arguments against the validity of the proof:
-Sagredo asks why the motion will not degenerate toward some other kind of curve. Since the vertical axis lies on a line extending from the center of the earth, and all bodies have a tendency toward the center of the earth, a projectile body should eventually head towards the center of the earth, which would cause a deviation from the parabolic path.
-The first of Simplicio's two objections is that the horizontal axis would continue further away from the center of the earth, meaning that the horizontal component of the motion would not remain equable over a great length of time.
-His second objection is that the impediment of the medium would effect both the horizontal component and vertical component of motion enough that an actual parabola would never be observed.[16]
Simplicio sums everything up nicely, saying, “All these difficulties make it highly improbable that anything demonstrated from such fickle assumptions can ever be verified in actual experiments.”[17]
So unless this proof is tested over a very small distance, in a medium with very low resistance, it is unlikely that experiment will confirm the result. Salviati isn't caught entirely off-guard by their questions. He responds, “I admit that the conclusions demonstrated in the abstract are altered in the concrete, and are so falsified that horizontal [motion] is not equable; nor does natural acceleration occur [exactly] in the ratio assumed; nor is the line of the projectile parabolic, and so on.”[18]
Salviati then attempts to justify why we can still admit abstract proofs when they do not ever come true in the sensible world, as well as explain what use they still have. He concludes that without the ability to abstract, we could not do science. Salviati says,
No firm science can be given of such events of heaviness, speed, and shape, which are variable in infinitely many ways. Hence to deal with such matters scientifically, it is necessary to abstract from them. We must find and demonstrate conclusions abstracted from the impediments, in order to make use of them in practice under those limitations that experience will teach us.[19]
We have not actually established a good definition of science up to this point, but Salviati may have given us one. Science is the necessary abstraction of physical phenomena in order to reach certain conclusions about physical phenomena. This definition fits with the conclusion in the last section; no matter how clever the method, a scientific proof is useless without experience.
But Salviati does not think this definition means that scientific proofs to have a perfect representation in the physical world. If an abstract proof does not fit exactly with practical experience, the method on which the abstract proof was built should not be thrown out. If the abstract proofs are not used, then nothing could be concluded. The resolution of Simplicio and Sagredo's issues would be no science at all.
One probably feels sympathy for Simplicio and Sagredo. Salviati has finally conceded a point to them, but if the point were fully conceded, they would have to reject any certainty.
In order to have a good method, the conclusions made in an abstract system, such as the one Galileo has constructed, must have application. Simplicio and Sagredo's position is too extreme; nature has so many variables that the mind cannot account for them all. If knowledge concerning nature is desired, the knowledge must be comprehensible. Salviati cannot entirely ignore Sagredo and Simplicio's objections and arbitrarily construct proofs; he must be always aware of Sagredo's warning that he is arguing ex suppositione. A Galilean proof is not a monument in the way a proof of pure mathematics might be; his proofs will lose relevance once they cease to explain nature correctly.
Simplicio and Sagredo do not fight back, but we may assume that their fears have not been put to rest; perhaps their faces register a quiet uneasiness, accompanied by some head-scratching. Looking at the dialogue surrounding Proposition 3.I, there are two major obstacles between Simplicio and Sagredo and the acceptance of this new method. The first is the technical gap which separates Salviati from the other two interlocutors. In order to accept any conclusions with this new method, the mathematical foundations of the method must be comprehended. The second obstacle is philosophic. Even if they can comprehend the mathematical foundations, will they be able to accept that the proof has experimental validity?
Breaking the Prism, Cutting the Stick
The first two days of Two New Sciences are a catalog of missteps in reasoning by Sagredo or Simplicio, followed by Salviati correcting their errors. The topics of discussion, especially in Day One, such as the existence of the void, the periodicity of a pendulum, and music, are far removed from the two new sciences promised at the beginning. What is Galileo showing us through this unessential dialogue and constant correcting of the lesser interlocutors?
Galileo may have used this unessential dialogue to prepare his readers, philosophically and technically, before they study his new sciences. The propositions of the first new science do not appear until Day Two (and we do not see the Author's treatises until Day Three), but the average reader would presumably want to read these proofs with the appropriate background. Not only an appropriate technical background–e.g. knowing that the volume of a solid increases in a three-halves ratio to its surface area–but a philosophical background as well. With the wrong philosophical foundation, the scientific frame of mind cannot be achieved, and the student of Galileo will be unable to practice his method appropriately.
The correct philosophical approach is covered in the discussion between the interlocutors in Proposition X of Day Two.[20] The question under discussion is fairly basic: If we are holding a stick with a hand at each end, does it matter where on the stick we try to break it across our knee? That is, would the same force be needed at any point along the stick, or would it become easier or harder to break the stick at different places? Sagredo, “at first glance,” says that you could use the same force at any point along the stick, and the stick will break. The reason he gives is logical–as you move along the stick whatever change in resistance occurs from lengthening the distance on one side would be offset by shortening the distance on the other side, so that the sum total of moments[21] is the same regardless of where the knee hits. Sagredo is wrong. The reason is complex, but here's a summary: When you bring your knee up to break the stick, the cohesion of the body will resist the force of the knee up to a certain point. Once the force from the knee is too great, the stick will break. The sections on either side of the stick's breaking point are, mathematically, two separate sticks. They do not have a total moment which is conserved at all points along the stick, as Sagredo reasoned. Salviati explains that if you attempt to break the stick closer to one side than the other, such as point F, the length of the shorter side will decrease in infinitum as the leg gets closer to that end of the stick. Since force x distance = moment, a decrease in the length DF will correspond to a decrease in the moment at D, assuming that force stays the same. To compensate, the hand applying the force on that side will need to increase the applied force in infinitum. The length AC can have an infinitely large length in comparison to DF. While this is happening, the hand on the long end of the stick (point E in the diagram) must continually decrease the applied force, in order to keep the stick from sliding off the leg. However, the length of the long end of the stick EF will never reach a 2:1 ratio with the length BC. This means that the moment at point E will never be more than two times as great as the moment at point B, assuming the applied force remains the same. Sagredo says that the total moment is conserved at any point along the stick, and Salviati shows that he's wrong. The least amount of total force is necessary when the knee strikes exactly in the middle.
Sagredo assumed that since the total length of the stick never changed, the total moment along the whole stick never changes. He failed to observe how the ratios would change from when the stick is broken at center point C, to any non-bisecting point F. Sagredo's did not take into account the subtleties that geometry presents.
Sagredo marvels at this conclusion,
What shall we say, Simplicio? Must we not confess that the power of geometry is the most potent instrument of all to sharpen the mind and dispose it to reason perfectly, and to speculate? Didn't Plato have good reason to want his pupils to be first well grounded in mathematics? I understood quite well the action of the lever, and how by increasing or reducing its length, the moment of its force and the resistance grew or diminished; yet for all that, I was mistaken in the solution of the present problem, and not a little, but infinitely.[22]
Sagredo had the background to reach the same conclusion as Salviati (as he indicated when he said, “I understood quite well the action of the lever”); he didn't lack a technical understanding. Rather, he was unable, when faced with Salviati's question, to appropriately apply this knowledge. He failed to think about the problem with the correct frame of mind. The difference between Sagredo and Salviati is that when Salviati is faced with a problem, he uses mathematical reasoning, while Sagredo can follow the reasoning, but not perform it.
This is the second time in the dialogue in that Plato comes up in reference to a mathematical education. The first was when Simplicio said he “should follow the advice of Plato and commence from mathematics.”[23] When Simplicio says this in Day One, the interlocutors are discussing the effect of the resisting medium on the downward acceleration of a body in proportion to its heaviness. Salviati says, “It cannot be doubted that in the descent of moveables these [irregularities] rub against the surrounding fluid and bring about retardation of speed, greater as the surface is larger, as is the case with smaller solids in comparison with large ones.”[24] The phrase “greater as the surface is larger,” confuses Simplicio. Why should smaller solids have a greater surface area? Salviati explains that in a smaller body, the surface area has a greater ratio to the corresponding volume than in a body with a very large volume. It makes sense if we think about it with cubes. Comparing a cube with surfaces 1x1x1 and one which is 2x2x2, the two surface areas will be in a 1:4 ratio (6:24), whereas the volumes are in a ratio of 1:8. Upon this explanation, Simplicio renounces his prior education in favor of Plato.
Another example of the failure to recognize geometrical subtlety can be drawn from Plato himself, in the interaction with the slave boy in Meno.[25] Socrates asks the slave boy how much the side of a square needs to be lengthened to double its area from four to eight. The slave boy promptly responds that the length of the line should be doubled. This answer is wrong, of course. If the line were doubled, the area would expand to sixteen feet. The slave boy's mistake seems analogous to the mistakes made by Simplicio and Sagredo above. In each case, their answers, though wrong, seemed obvious. The right answer is one which requires further reasoning, but when the correct answer is reached, it is so clear that it is hard to see how one made the mistake in the first place. Socrates and Salviati, in contrast to Simplicio, Sagredo, and the slave boy, can see past the obvious answer to the correct answer.
Salviati acknowledges the importance of this trait early in Day One: “Here you and Simplicio must note how conclusions that are true may seem improbable at a first glance, and yet when only some small thing is pointed out, they cast off their concealing cloaks and, thus naked and simple, gladly show off their secrets.”[26] The good natural philosopher, a person who can consistently reach correct conclusions about nature in the way Salviati does, must possess excellent geometrical reasoning, more than anything else. Sagredo's failure to solve the stick-breaking problem shows that a good scientist is not just a storehouse of technical knowledge, but possesses a sharp eye for reasoning as well.
After the stick-breaking problem, Simplicio contrasts geometrical reasoning with logical reasoning, saying, “Truly I begin to understand that although logic is a very excellent instrument to govern our reasoning, it does not compare with the sharpness of geometry in awakening the mind to discovery.”[27] The distinction made between mathematical and logical reasoning describes Sagredo's misstep in the case of the knee breaking the stick. When Salviati asked whether the same force should serve at all places, he says that it should, because if one side is lengthened, the other must be shortened accordingly, meaning that the added moment on one side would be proportionally decreased on the other side, resulting in the same total. At the time, Sagredo felt like he was giving the logical, correct answer.
The conclusion Sagredo comes to seems more intuitive in the same way the Fancy seemed like the more intuitive method of finding the law of uniform acceleration or doubling the length of the side seemed like the right answer for the slave boy. Sagredo is never at a loss in technically understanding Salviati's proofs, but somehow he can't apply Salviati's reasoning when forced to solve a problem on his own. He seeks the intuitive, or immediately obvious solution to the problem, rather than the geometrical solution, and that is the issue.
The philosophical approach one needs study nature is a mathematical approach, not a logical approach. With the multiple references to Plato, it seems that Galileo would go even farther and say that we should approach nature as Platonists, rather than Aristotelians.
Despite the recognition of the superiority of mathematical reasoning by Sagredo and Simplicio in Day Two, their revelry quickly ends when faced with another problem.[28] The interlocutors want to figure out if a rectangular prism can be cut in such a way that it will resist breakage equally well at all points while removing the greatest amount of material. Salviati has recognized that a rectangular prism (such as the one pictured) when cut diagonally along the line FB, will resist being broken less as the fulcrum moves closer to point B. As Salviati says, Hence the resistance of part OCB to being broken [off] at C is as much less than the resistance of all DAB to being broken at A as the length CB is less than AB. Thus we have taken away from the beam or prism DB a part, in fact one-half, by cutting it diagonally, leaving the wedge or triangular prism FBA; and these two solids [the rectangular prism and the triangular prism] are of contrary condition, the former being more resistant the more it is shortened [in the direction of B], and the latter losing robustness as it is shortened.[29]
According to Salviati, in the rectangular prism, shortening the length improves the resistance. In the triangular prism ABF which is produced from the diagonal cut, so much material is removed that this is reversed – the prism will become less resistant if made shorter.[30]
From this reasoning, Salviati asks what cut would remove all superfluous material, while the remaining solid would be equally resistant to breaking at all points. Simplicio, energized from his recent transformation, takes on the challenge. Simplicio says in his answer:
It seems to me that this should be an easy task. By cutting the prism diagonally and taking away half, the shape that remains has its nature contrary to that of the entire prism, in such a way that wherever the latter gained strength, the former lost as much. So I believe that we should take the middle path; that is, by taking only one-half of the half, or one-quarter part of the whole, the remaining figures will neither gain nor lose robustness at any of those places at which the other two figures had equal losses and gains.[31]
His wording and reasoning closely resemble Sagredo's in the stick-breaking problem–“wherever the latter gained strength, the former lost as much”–and his folly is understandable. But in this case, he fails to realize that any straight line will be the incorrect answer. The straight line will never work because the total resistance is not balanced evenly throughout the prism; there will always be one side which has an excess, while the other side lacks material. When thinking about the problem with the correct philosophical approach, one should come to realize that a different straight line will not be the answer; it would only move the fulcrum point to another spot along the prism. A different shape has to be the answer, and Salviati reveals that the parabola is correct (as we know from Day Four, a shape with which Simplicio is not familiar). Simplicio is able to recognize that geometrical reasoning is the correct method, but, like Sagredo, he gives the obvious, incorrect answer when pressed.
While some may laugh at Simplicio for these missteps, one can still have sympathy for him. When he is forced to solve a problem, he does not revert to the logical reasoning because he is lazy, but because of his lack of training. His problem is one step below Sagredo's. Sagredo understands the technical, mathematical side, but lacks the mindset to think mathematically when pressed. Simplicio may have a clever and incisive imagination, but we will never know that, since his head is not filled with the background to put this imagination to use. And so, regardless of how simple and fruitful Galileo's method of mathematical reasoning is, its technical requirement makes it unapproachable for Simplicio. Simplicio seems right when he says that logic “governs our reasoning”; it is likely that most would have first answered the question in the ways that Sagredo and Simplicio did. Yet there still may exist a whole group of clever reasoners whose reasoning powers are obscured by their insufficient technical understanding.
“You have not hit the target, Simplicio.” Salviati quickly cuts down Simplicio's thesis with these words. As a result, Simplicio does not speak again for the rest of Day Two. Yet presumably he remains present at the discussion. How is the dialogue staged at this point? Where does Simplicio go? What emotions does his face conjure? Only a few moments ago, he had reached one of his greatest breakthroughs; Salviati had just successfully convinced him, through Sagredo's mistake, of the superiority of mathematical reasoning. So imagine his dejection when he is quickly asked to solve another problem, and he's wrong. Not only is he wrong, but he's wrong because the correct answer was never available to him; he does not know anything about parabolae.
Simplicio is not seeking to deny the superiority of Galileo's method. What slows him down is his education. His excellent knowledge of Aristotle leads him to think about physical situations in Aristotelian terms. Through Day One, Salviati tries to break Simplicio of this prejudice, and Simplicio acknowledges the superiority of an education rooted in mathematics, as discussed above. There is no reason to think that Simplicio does not seek to be a model student of nature in the Galilean mold. Yet it does not appear that he is capable of being so.
Catching Up with the Racehorse
How should someone like Simplicio–the person who desires correct understanding, but who lacks the training–proceed? To explore this question we are briefly going to look elsewhere at The Assayer, a letter refuting critiques (primarily Scholastic) of Galileo's conclusions concerning the nature of comets.
The central thesis of The Assayer is that the study of nature should not proceed according to received wisdom from previous authors (see: Aristotle) but that we should turn to our own experience as a confirmation of natural truths. This quote is emblematic of the theme throughout the letter,
Sarsi[32] goes on to say that since this experience of Aristotle's has failed to convince us, many other great men also have written things of the same sort. To this I reply that if in order to refute Aristotle's statement we are obliged to represent that no other men have believed it, then nobody on earth can ever refute it, since nothing can make those who have believed it not believe it. But it is news to me that any man would actually put the testimony of writers ahead of what experience shows him.[33]
It is easy to support the conclusion that our own experience should be the final arbiter when thinking about nature, instead of another author. It is hard to see, though, how this quote fits with this passage on the following page,
But even in conclusions which can be known by reasoning, I say that the testimony of many has little more value than that of few, since the number of people who reason well in complicated matters is much smaller than that of those who reason badly. If reasoning were like hauling I should agree that several reasoners would be worth more than one, just as several horses can haul more sacks of grain than one can. But reasoning is like racing and not like hauling, and a single Arabian steed can outrun a hundred plowhorses.[34]
This conclusion is also obvious – we don't need to read much of the Two New Sciences to figure out that Salviati is the Arabian steed of the three. But for an initially poor reasoner like Simplicio, it is difficult to know when to trust a better reasoner, such as in the proof of projectile motion, instead of trusting his own reasoning. On the one hand, individual experience can be more correct than the accumulated wisdom of the past. On the other hand, certain people are naturally better reasoners.
In Two New Sciences, all three interlocutors can experience equally well. There is no indication that they observe phenomena in wholly different ways. Yet anyone reading Two New Sciences will quickly realize that there is a hierarchy in the group; Salviati consistently leads both the reader and the other interlocutors to conclusions that seem clear and obviously correct. What makes Salviati, or other “Arabian steeds” capable of better reasoning? Secondly, can poor reasoners, like Sagredo and Simplicio, become better reasoners?
Along with good technical knowledge, what distinguishes Salviati as a thinker are the two things recognized in the quotes from The Assayer above: 1) He trusts his own experience over the accounts of previous writers (he lacks prejudice), and 2) He reasons well, meaning that he can use geometrical knowledge to take a physical problem and abstract from it in order to make it solvable.
At the beginning of Two New Sciences, Sagredo says:
And since I am by nature curious, I frequent the place for my own diversion and to watch the activity of those whom we call “key men” [Proti] by reason of a certain preeminence that they have over the rest of the workmen. Talking with them has helped me many times in the investigation of the reason for effects that are not only remarkable, but also abstruse, and almost unthinkable. I have sometimes been thrown into confusion and have despaired of understanding how some things can happen that are shown to be true by my own eyes, things remote from any conception of mine.[35]
In order to be a strong reasoner, one must distrust one's immediate conceptions, and use the certainty of geometry to determine truth. Sagredo's quote conveys how difficult it is to do this in practice.
Take the problem of naturally accelerated motion, which is discussed at length in Day One[36] and is the central concern of Day Three. The prevailing Aristotelian view was that bodies fell in proportion to their heaviness. Galileo, on the other hand, said there is a simple law of uniform acceleration which governs all downward motion, and that any perceived difference in times of fall is due to the effect of an external medium in proportion to the size of a body. We moderns now rather easily accept the Galilean picture of things, so why would many generations of people easily agree with Aristotle's view, even though their experience wasn't much different from Galileo's or ours?
Many bodies do fall at obviously different speeds; a feather and a cannonball will have significantly different times of descent from the same height. The Aristotelian view–that bodies fall in proportion to their heaviness–probably emerged from this type of observation. As Salviati shows, Aristotle's law falls apart when taken to its logical extreme. A tennis ball is a Lilliputian in comparison to a normal cannonball, yet if the two bodies fall over the same distance, the difference in speeds will hardly reflect the same ratio as their difference in weights. If the cannonball were a thousand pounds and the tennis ball one pound, according to the law of fall in proportion to weight, the distance covered by a cannonball in one second would be the same distance covered by a ping-pong ball in a thousand seconds. Clearly, this vast a difference in speeds will not be experienced.[37]
So what does account for these perceived differences? Salviati attributes them to the resistance of the medium. He reasons that in water, a more resistant medium, bodies of different densities will have a greater difference in their speeds of fall than their difference of speed in air. Theoretically, there could exist a medium with no resistance–a void–where all bodies fall with the same speed. It is concluded, then, that the medium causes any difference in times of fall, something external to the body. And furthermore, the falling time of all bodies is inherently governed by the same law.
Initially, it would seem improbable to say that all bodies naturally fall with the same speed; our everyday experience of falling bodies rarely shows two bodies hitting the falling in tandem. The natural response, and the Aristotelian response, was to consider the different falling times as a property of each individual body. Salviati was able to remove this “concealing cloak,”[38] and show, through quite simple reasoning, that this conclusion has absurd implications, and any differences we perceive are due to the medium. Once he has made this conclusion, it is hard to see how anyone might conclude otherwise.
So where did Salviati succeed and Simplicio and Sagredo fail? They all had experience of falling bodies–there wasn't even need for a special instrument like a telescope. What separated Salviati, in this case and many others, was his capability for abstraction. He took his general experience of bodies in air and bodies in water and perceived the different ways similar bodies behaved in those media. In more resistant media, the bodies grow further apart more quickly, and as the resistance becomes less, the bodies which were far apart in a dense medium fall at more similar speeds. His reasoning is really a thought experiment; he understands nature well enough to understand rules for how bodies should act, even when he does not directly experience the action.
This capacity for abstraction appears in the examples cited above. In Proposition 2.I the author defined downward accelerated motion in terms of horizontal uniform motion, even though the two types of motion don’t seem to relate when seen in nature. The proof of projectile motion rests on the ability to take two previous proofs and think of them geometrically, that is, abstractly, to derive the parabola.
A lack of prejudice is the other key to Salviati's skill. One of Simplicio's difficulties with Salviati's approach to the law of uniform acceleration was that it required a void. Aristotle says that the void does not exist.[39] Simplicio, who “puts the testimony of writers ahead of what experience shows him,”[40] is unable to entertain Salviati’s argument because he holds on to Aristotle’s belief. Even in Two New Sciences, Salviati does not experience the void, but Simplicio rejects Salviati's argument because it implies the void. Even though the argument is reasonable, Simplicio's prejudice prevents him from accepting Salviati's conclusion, because it goes against something previously written, which Simplicio believed.
When the previous section closed, the implication was that Simplicio will continue being Simplicio. Even after acknowledging that the method of geometrical reasoning was the more fruitful method, when faced with the problem of how to cut the prism only a little while later, he was unable to solve it correctly. Unless Simplicio can overcome his technical deficiencies, he will never improve break out of last place in the proverbial horse-race.
Still, Two New Sciences covers a very short time. If the interlocutors met in a year, Simplicio could return confident and newly educated. The dialogue may simply show too small a span of time to see Simplicio flourish. It is possible to acquire the technical knowledge needed for understanding these two new sciences. Sagredo showed this in Day Four; once he had the proofs from Apollonius, he seemed to understand Proposition 3.I.
Say Simplicio did get rid of his technical difficulties overnight, and pulls neck-and-neck with Sagredo. What chance do they now have of catching Salviati? That is, can Simplicio and Sagredo learn the correct philosophical mindset? It appears unlikely that they will be able to.
The necessary geometrical mindset is not a byproduct of technical knowledge; a mindset cannot be studied in the same way as a mathematical proof. Rather, good reasoning shows itself only when the problem comes forward. Where should the knee hit the stick? How should the prism be cut? It is only revealed that Simplicio and Sagredo have the wrong philosophical mindset when they attempt to solve a problem.
If someone like Simplicio tries to improve his reasoning powers, a crucial trap must be avoided. People might reason poorly because they are too prejudiced in favor of a certain author. If this is the case, the solution is easy – follow Galileo's advice to trust one's own experience, and the reasoning will follow. It seems equally likely, though, that the cause and effect chain is reversed. That is, because someone recognizes his poor reasoning capabilities, he trusts in other authors because he doubts his ability to interpret his own experience. This interpretation is fatal for the student hoping to improve his reasoning; if he trusts others because he doubts himself, the future is bleak.
Making Sense of Science
Think back to the moment in Day Two, when Simplicio fails to solve the problem of cutting the prism. He has acknowledged the superiority of the method of mathematical reasoning, but has failed to apply it appropriately. This disconnect seems to exist for Sagredo as well. So how can someone acknowledge the superiority of this method without being able to practice it well? What is their recourse?
In Sagredo's first speech of the dialogue, he tells the others that he frequents the Venetian Arsenal (where the dialogue takes place) in order to learn from certain “key men.”[41] Sagredo says that talking with them helps him solve problems that were originally obscure. Salviati seems to be a key man, given how he helps the other interlocutors throughout the dialogue. Sagredo's acknowledgement of key men–those who reason better than others–is encouraging in one way, since it is good to know that this type of knowledge can be explained to less capable reasoners with success. Yet without key men around, Sagredo and Simplicio would be unable to make discoveries for themselves. They are slaves to the strongest reasoners.
Galileo shows in The Assayer that he has little patience for people like Sarsi, but he fails both in The Assayer and Two New Sciences really move beyond his anger and ask: If only a few people are truly able reasoners, what should be done about those who aren't? How is it that someone, like Simplicio, could fully accept this method without understanding it – or as he said in Day Four, believe without comprehending? Would he be content in that position? Back in the Fancy, Simplicio “takes more pleasure” in Sagredo's presentation than Salviati's. As was noted then, Simplicio sees his choice as a matter of taste, not one of mathematical or philosophical rigor. We are only able to accept a system of thought when it makes sense to us, that is, when it fits with the taste of our intellect.
In The Assayer, Galileo says “the number of people who reason well in complicated matters is much smaller than that of those who reason badly.”[42] In other words, there are few key men amongst the multitude. If this is true, then how should the multitude believe in a proof which does not make sense to them? If the criterion for acceptance of Galileo's proofs is that they “make sense,” many will not accept unfounded belief over understanding, as Simplicio did when he rejected the Author's proof of uniform acceleration on the grounds that it was obscure. If most people are bad reasoners, then most people will be convinced by bad reasoning.
At crucial junctures – the law of uniform acceleration and proof of projectile motion, for instance – Simplicio never reaches the understanding that Salviati has. If Salviati is always correcting the reasoning of the lesser interlocutors, then the students will doubt their own abilities more, and their courage will waver.
That said, comprehension can be acquired in degrees. Salviati comprehends better than Sagredo, and Sagredo better than Simplicio, and there are many other levels of comprehension in between which we can imagine. By framing Two New Sciences as a dialogue, Galileo shows that he acknowledges these varying levels of comprehension. Passages like Sagredo's Fancy show that Galileo wants to make a connection with readers who might not relate to Salviati.
Yet there are points when these interlocutors, with their different levels of comprehension, fail to relate. Regardless of Galileo's best efforts, the knowledge he proselytizes will remain obscure to some.
A comparison was made earlier in the paper to Plato's Meno, and another comparison is appropriate here, if only for contrast. The scientific knowledge presented in Two New Sciences does not seem recollectable – in the case of the Fancy, Sagredo’s recourse was to create a new proof, rather than trying to understand the one just presented.
Yet Salviati still plays a sort of midwife. If he weren't there, the lesser interlocutors would be lost in the wilderness. Over time and with enough study, could the lesser interlocutors take on problems of their own? After reading Two New Sciences, the intellectual distance between the three seems largely fixed. Any improvement will not be due to an improvement in reasoning as much as it will be an improvement in the “ineffable qualities” alluded to in the introduction. Simplicio and Sagredo must see if they have the confidence to take on the challenge of acquiring this knowledge, and whether this is something worth knowing.
If the multitude fails to take this challenge, the consequences are severe. If most people are poor reasoners who reject their own experience in favor of another person’s words, what’s the use does this knowledge have? And what's the danger of knowledge that few people understand?
[1] All quotations in Two New Sciences are taken from the Stillman Drake translation, published by Wall & Emerson in 1989. Citations will abbreviate the title of the book, followed by the marginal number. This first citation reads: TNS 210
[2] There are three Latin treatises, written by Galileo himself, that appear in the dialogue. For ease of reference, when I refer to a proposition in these treatises, the first Arabic numeral will refer to the treatise in the order in which they appear: 1) Equable Motion 2) Naturally Accelerated Motion 3) Projectile Motion. The Roman numeral corresponds to the number of the proposition within that treatise.
[3] All images are taken from an electronic copy of Dialogues Concerning Two New Sciences, translated by Antonio Favaro, published by Macmillan in 1914.
[4] TNS 212
[5] Pictured at the side of this page and on pgs. 167 and 168 of Drake's translation.
[6] The enunciation reads, “The time in which a certain space is traversed by a movable in uniformly accelerated movement from rest is equal to the time in which the same space would be traversed by the same movable carried in uniform motion whose degree of speed is one-half the maximum and final degree of speed of the previous, uniformly accelerated, motion. (TNS 208)
[7] TNS 210
[8] TNS 210
[9] The enunciation reads: “Of movements through the same space at unequal speeds, the times and speeds are inversely proportional.” TNS 193
[10] TNS 212
[11] TNS 269
[12] TNS 270 [emphasis mine]
[13] TNS 269
[14] TNS 134
[15] This premise can be derived from many places in Galileo's writings, but it is stated most clearly in the final paragraph of the message from the Printer to the Reader, on pg. 8 of TNS.
[16] This comes up with no mention of their discussion on the resistance of the medium in Day One. See TNS 116.
[17] TNS 274
[18] TNS 274
[19] TNS 276
[20] TNS 172-176
[21] In the Glossary to TNS, Drake defines “moment”, a translation of the Italian momento, as follows: “Static moment is conceived by Galileo as the effective downward tendency of a weight acting through a lever arm, and he treats it as the product of weight and distance, in that any one change is exactly compensated by an inversely proportional change in the other.” See pg. Xli of the Drake translation.
[22] TNS 175
[23] See TNS 133-135
[24] TNS 132
[25] Meno 82B-85C
[26] TNS 52
[27] TNS 175
[28] TNS 177-185
[29] TNS 179
[30] See the second paragraph of Salviati’s speech at TNS 178-179. Salviati says there, “And these two solids are of contrary condition, the former being more resistant the more it is shortened [in the direction of B], and the latter losing robustness as it is shortened.”
[31] TNS 179
[32] “Sarsi” is the villain of The Assayer. It is the assumed name behind which a person criticized the findings of Galileo, driving him to write this letter.
[33] All quotes from The Assayer refer to the excerpted version in Discoveries and Opinions of Galileo, by Stillman Drake, published by Doubleday, 1957. Unfortunately, there are no marginal numbers. This quote appears on pg. 270
[34] The Assayer, pg. 271
[35] TNS 49
[36] See TNS 107-135
[37] This reasoning is used by Salviati at TNS 106.
[38] TNS 52
[39] See Book 4, Chapter 8 of Physics
[40] The Assayer, pg. 270
[41] TNS 49
[42] The Assayer, pg. 271