Roger Bissell

mathematics essentially deductive?

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Mike,

Basically, my argument has been in agreement with your viewpoint, if you have followed it through the thread. I believe that mathematics should be a tool for understanding reality and as such it does depend upon inductive reasoning heavily. However, some pure mathematicians simply take a set of rules, which may or may not have a relationship to reality, and they play logical games with them. Some people who claim that mathematics is purely deductive have this approach in mind. Well, if that were all math was good for, then it is simply a game like chess and it actually has no explanatory capability with respect to the real world.

When and where math gives insight to reality, it does so with a heavy dose of induction. Those who say the contrary are generally trying to remove the mathematician from reality and leave that as the exclusive field of physics. Personally, I like applied mathematics and those who practice it. There are some mathematicians who pride themselves in being pure mathematicians and they actually act as though thinking of math as a tool is demeaning to mathematics. I do not share this view, though I recognize that pure mathematicians can play games that are pretty divorced from reality if they really want to.

Charles

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Mike,

Basically, my argument has been in agreement with your viewpoint, if you have followed it through the thread. I believe that mathematics should be a tool for understanding reality and as such it does depend upon inductive reasoning heavily. However, some pure mathematicians simply take a set of rules, which may or may not have a relationship to reality, and they play logical games with them. Some people who claim that mathematics is purely deductive have this approach in mind. Well, if that were all math was good for, then it is simply a game like chess and it actually has no explanatory capability with respect to the real world.

Could you be specific? Which mathematicians do you have in mind who do that?

And __who__ claims that mathematics is purely deductive (other than non-mathematicians

or those who don't care about philosophy)?

When and where math gives insight to reality, it does so with a heavy dose of induction. Those who say the contrary are generally trying to remove the mathematician from reality and leave that as the exclusive field of physics.

Why __physics__, specifically? Are you claiming physics is the only science

to which mathematics is applicable? What about statistics applied to psychology

or to economics? What about applications to financial markets?

Personally, I like applied mathematics and those who practice it. There are some mathematicians who pride themselves in being pure mathematicians and they actually act as though thinking of math as a tool is demeaning to mathematics. I do not share this view, though I recognize that pure mathematicians can play games that are pretty divorced from reality if they really want to.

Is it not true that even in its "pure" form, mathematics relies heavily on

induction for concept-formation? -- Mike Hardy

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I think one example of a mathematician championing "pure mathematics" and who was offended by the idea of practical applications being the yardstick of an idea's value is G. H. Hardy, the great early 20th century British mathematician. He was a giant in the field of number theory, co-writing a classic text in that field. A couple of quotes from "A Mathematician's Apology" clearly portray his bias toward theory:

I am interested in mathematics only as a creative art.

Pure mathematics is on the whole distinctly more useful than applied For what is useful above all is technique, and mathematical technique is taught mainly through pure mathematics.

However, I don't believe that Hardy championed "pure deduction" in mathematics, i.e., starting out with arbitrary postulates and deducing logical consequences. He certainly appreciated rigor and the need to nail down one's intuitive insights and hunches, but his approach was first and foremost inductive and/or intuitive, rather than deductive and logical.

Peikoff, in his induction lectures, claimed that mathematics was different from the other sciences, in that it was essentially or primarily deductive. He perhaps was thinking of geometry. But even there, other Objectivists (notably, Pat Corvini) have convincingly made the opposite case, that mathematics is essentially inductive. Not to denigrate the value or usefulness of deduction to validate and firmly situate mathematical conclusions within a structure of more general and more specific conclusions -- but to stress that mathematical discovery is primarily inductive, not deductive.

Let me give a "simple" example that I think will illustrate this point.

Suppose I wanted to demonstrate to you, or help you to discover, that the expansion of (x + 1)(x - 1) is always equal to x^2 - 1, no matter what the value of x. [x^2 stands for x-squared or x times x, given the limitations of this site's word-processing capabilities.]

I could just have you apply, deductively, the polynomial multiplication rules thusly: x^2 + 1x - 1x -1^2, and you would see that the two middle terms always drop out, leaving x^2 - 1. This is a pretty quick way to demonstrate the expansion of (x + 1)(x - 1) is x^2 - 1, no doubt about it. But what use would there be in having seen that? Apparently, none, other than having learned a formula for use in simple problems such as: what is the area of a rectangle whose length is 2 units longer than its width? Typically we would multiply w (width) times (w + 2) (length) and get w^2 + 2w. For instance: if width is 9 and length is 11, then the area, expressed in terms of the width is 9 x (9 + 2), which is (9 x 9) + (9 x 2) = 81 + 18 = 99. This is really the long way around, when we could instead just plug in the numbers 9 and 11 to the w times l formula and get 99 simply and directly. Not too useful, to be sure.

However, let's take a different, more interesting approach. Suppose instead we note that there is another quantity x, in between the length and width, and in relation to which length is x + 1 and width is x - 1, and that the area of the rectangle is (x + 1) times (x - 1) or x^2 minus 1. We can generalize: the area of a rectangle with length 2 units longer than the width is one less than the square of the number that is ONE unit longer than the width. For instance, 7 x 5 is one less than 6 x 6, 9 x 7 is one less than 8 x 8, 21 x 19 is one less than 20 x 20. Now, we can see some usefulness to using expansions in multiplication, even of numbers, rather than just letters. Suppose you want to multiply 99 times 101 (aka find the area of a rectangle 99 units wide and 101 units long). Instead of "simply" multiplying these numbers (in your head, if you're really smart), you can square the integer in between them and subtract 1 -- 100^2 - 1 = 10,000 - 1 = 9999. Now, that, I would say, is a benefit, a useful application of expansions. It falls in the category of what some call "speed math." And it is just one of numerous examples.

The catch, though, is that you have to have the insight, the abstract capacity to "see" that this squaring a number then subtracting 1 is the same as the multiplying the next larger number by the next smaller number -- and that the former is EASIER, that it takes less time and effort, than the latter. My belief, after years of pondering this issue, is that "seeing" such economies or useful applications is much more difficult when you approach a particular idea deductively than when you approach it inductively. That is, it's easier to "see" the payoff, the practical value, of ideas when you grasp them inductively.

I'll try to illustrate this with the same example, the (x + 1) times (x - 1) expansion.

Suppose we lay out several columns of numbers:

x -----x + 1-----x - 1-----(x + 1)(x - 1)-----x^2

1--------2---------0---------------0--------------1

2--------3---------1---------------3--------------4

3--------4---------2---------------8--------------9

4--------5---------3--------------15------------16

5--------6---------4--------------24------------25

6--------7---------5--------------35------------36

7--------8---------6--------------48------------49

By "inspection," we can see "with the overwhelming clarity of direct perception" that there is an inexorably repeating pattern tying these columns together. The square of a given integer, x, is always going to be 1 greater than the product of the next larger and next smaller integer. In algebraic terms: x^1 = (x + 1)(x - 1) + 1, or (x + 1)(x - 1) = x^2 - 1. Or, so it appears. We can quickly verify that this is so by means of the deductive technique shown above. But before we do that, we note that IF it is a true generalization, there is a very important and useful spin-off: if we want to multiply any two integers that differ by 2, all we need do is square the integer between them and subtract 1. Many people, for instance, know that 12 x 12 = 144. Now they can easily multiply 11 x 13 in their heads: 144 - 1 = 143. (For other examples, see above or make up your own.)

This is the payoff of induction, I think. It takes talent, if not genius, to grasp "speed math" from the deductive route. But with induction, grasping "speed math" takes no particular talent, just straightforward generalization and the realization of what one has been doing, entry by entry in the above table of numbers, namely, displaying and comparing the square of integers with the product of the next greater and next smaller number -- and than having the flash of realization that this can be a useful, time-saving technique.

This is how I wish students were taught algebra, with the roll-up-your-sleeves, generalize from concrete instances kind of work. This, in general, is how creative discovery occurs in math. (Some people, such as the incredible Indian genius Ramanujan, who was sponsored by Hardy, are able to do it in their heads. I can only shake my own in awe at such cerebral power.) I myself have made some rather interesting, if obscure discoveries of my own in this manner -- such as a new way to generate Pythagorean triples (integers of the form x^2 + y^2 = z^2) -- and I KNOW that those discoveries would never have occurred to me if I had somehow managed to stumble on their mathematical expressions by the deductive route. Perhaps that's just me -- but various theorists including Polya (whom I recommend highly) also stress the inductive, creative side of mathematics, especially for young (high school and college) students.

One other interesting facet of this general issue has to do with Einstein's E = mc^2, which I delve into, in a mostly light-hearted way, in my essay, "How the Martians Discovered Algebra."

Best to all,

REB

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Mike,

You work so hard to disagree with me!

Pure mathematicians: A number theory professor at George Washington University, the wife of a physicist I have worked with. A professor I had at Brown from the Mathematics Dept. in the late 60s. These are the ones I have had personal interaction with. Usually, I just read books on math these days and usually to learn a particular technique so I can do a calculation.

Of course I accept the use of math in economics and psychology and farming and other fields.

Mathematics should rely heavily on induction for concept formation and in practice it usually does. But sometimes people have Platonic notions of the Ideal which they try very hard to divorce from reality. Sometimes one can take a valid concept arrived at from induction on observations of reality and couple it with arbitrary and unreal ideas and still have a kind of mathematics which will not explain anything in reality. If the realistic elements were thought out by other mathematicians, then a particular mathematician might take a set of valid concepts and wed them to further rules and concepts that have no relation to reality and deduce a series of results which will have no relation to reality. He may claim he is doing math.

Look, I made it very clear earlier in this thread that mathematics as a rational field of investigation is very inductive.

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Mike,

You work so hard to disagree with me!

I'm NOT working to disagree with you.

But look: G.H. Hardy and his ilk did NOT believe in

_arbitrary_ axioms, even if they said they did. Believing

in doing research with no applications is not at all the

same thing as believing in arbitrary axioms. And if they

used the word "arbitrary" they probably meant something

like "arbitrary" if one were to judge ONLY via the canons

of deductive logic, and were just callously disregarding

philosophical distinctions. -- Mike Hardy

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But look: G.H. Hardy and his ilk did NOT believe in

_arbitrary_ axioms, even if they said they did. Believing

in doing research with no applications is not at all the

same thing as believing in arbitrary axioms. And if they

used the word "arbitrary" they probably meant something

like "arbitrary" if one were to judge ONLY via the canons

of deductive logic, and were just callously disregarding

philosophical distinctions. -- Mike Hardy

I should also add that he was wrong to think that his work

would never make any difference to "the amenity of the world."

But it he'd used arbitrary axioms, then that would have been

right. -- Mike Hardy

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Stephen,

That's one hell of a story about you and Objectivity. That's concrete proof that love of productive work is life enhancing.

I am interested in receiving the essays. All of them are extremely high-quality. What do I do?

Michael

Today I am happy to announce the site Objectivity Archive. Its address is www.objectivity-archive.com. This site is an archive and library of Objectivity, now freely open to all readers and researchers.

Objectivity is a journal of metaphysics, epistemology, and theory of value informed by modern science. It consists of two volumes, each with six issues. It was a hardcopy journal, for subscribers, published from 1990 to 1998. Its authors were both professional academics and independent scholars.

In addition to the complete, exactly replicated text of Objectivity, the Archive site offers additional helpful features such as ABSTRACTS for all the main essays and a SUBJECT INDEX and NAME INDEX for the entire 1770 pages of the journal.

Edited by Stephen Boydstun

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Phil Coates -- and whoever else wants to weigh in on this -- in your considerable experience of hearing Rand and Peikoff (and others) lecture about induction, deduction, and mathematics, do you recall if any of them ever conceded a significant role in mathematics for induction? I just finished listening to the Understanding Objectivism lectures, and Peikoff therein talked about mathematics being atypical (epistemologically) because it was deductive, whereas the typical pattern of human knowledge acquisition was by means of induction. Has he modified this view since 1983?

Are you aware of any different position on induction in re mathematics from the people who have published in The Intellectual Activist?

REB

Which mathematics? There are purely formal system of mathematics that are devoid of empirical content and there are mathematical theories which were specifically formulated to deal with the dynamics of motion in the real world. For example; calculus and differential equations.

Mathematics done rigorously involves proving theorems from postulates. That is deductive. But sometimes deduction is guided by intuition and real world examples and hints. That is inductive.

LP's grasp of mathematics is rather limited.

Ba'al Chatzaf

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From the ETHICS thread “Are There Moral Standards?”

Michael (MSK) [#100 – 1/24/09]

“Interestingly enough, both philosophy and science operate with axioms.”

Peter (Dragonfly) [#102 – 1/24/09]

“Only mathematics uses axioms, science does not.”

Michael [#105 – 1/25/09]

“Can you show me some science, any science at all, that does not include math as an essential component?”

Peter [#106 – 1/25/09]

“There are physical theories and physical laws, but no physical axioms. That those theories make use of mathematics which in their turn are based on a set of axioms is not relevant. For the physicist the mathematics is just a convenient tool and in general they don't bother too much about the fundamentals. A good example is the Dirac delta function, invented by a physicist. It's a very useful tool, but it didn't have a real mathematical basis (in fact it isn't a function at all) until the mathematician Laurent Schwartz developed his theory of distributions in which the Dirac 'function' can be rigorously defined. For the physicists it doesn't make a difference, the axioms of distribution theory are not essential to the physical theories. There are more examples of (quasi-)mathematical tools invented by physicists that only later got a solid foundation by the work of mathematicians.”

Bob (Baal Chatzaf) [#111 – 1/25/09]

“All of quantitative science is grounded on assumptions. But they are not considered (by scientists) to be self-evident truth. All sciences have to have a starting point, but the grundlagen need not be axiomatic.”

Thomas (General Semantic) [#116 – 1/25/09]

“There is a slight problem with the statement [‘Only mathematics uses axioms, science does not’], it should read ‘Only pure mathematics uses axioms, science does not’. . . . When a scientist uses any kind of mathematics he is attaching a physical meaning to the symbols whereas the mathematician is not. This is a huge difference. When we attach a physical meaning then we get to apply some measure of 'truth' or 'false' to the relation but in pure mathematics we are only concerned with consistency. So science utilizes mathematics (your use of 'include' is somewhat ambiguous) and sometimes it even precedes mathematics as the above example illustrates, but science is a fundamentally different activity.”

Emmanuel (EDonate) [#126 – 1/25/09]

“Yes, this is true however when the scientist adopts what was a 'pure math' model he expects/experiments with it until he is satisfied that the particular 'consistent' 'pure' math model he thought was appropriate properly resembles reality. Once the model that approximates or nails the phenomenon is discovered or developed it would be improper to say that mathematics was not included or utilized in the discovery and that mathematics was not fundamentally necessary to its discovery.”

Thomas [#133 – 1/26/09]

“I think the difference between pure mathematics and applied mathematics absolutely crucial. Do you consider pure mathematics knowledge? Knowledge of what? About the only thing one could say pure mathematics is knowledge of is possible relations that we can discern. So on one hand we have all known possible relations (mathematics) and all known empirical relations (everything else). Huge, important difference.”

Emmanuel [#134 – 1/27/09]

“I think pure mathematics is very important. How long does a pure mathematics problem stay 'pure' anyway? Some of the most complex algebra problems out there started out as pure math problems and are now being used in computer science. Pure mathematics models sometimes show parallels to applied mathematics models and tell you exactly what to expect. The line you're trying to draw is just not solid enough to be valid. Too many things move from the pure side to the applied side.”

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

See also:

“Mathematics in Science” by Stephen

http://www.objectivistliving.com/forums/in...?showtopic=6263

http://www.objectivistliving.com/forums/in...amp;#entry61242

http://www.objectivistliving.com/forums/in...amp;#entry61768

http://www.objectivistliving.com/forums/in...amp;#entry62439

Grounding Concepts: An Empirical Basis for Arithmetic Knowledge by C. S. Jenkins

http://rebirthofreason.com/Spirit/Books/232.shtml

Naturalism in Mathematics by Penelope Maddy

http://www.amazon.com/gp/reader/0198250754..._pt#reader-link

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

In Objectivity

Axioms; Laws of Physics as V1N5 82–83, 87, 90–91, V1N6 181–83; Mathematical V1N2 6–8, 12–14, 18–20, 22–23, V1N5 80–81, V1N6 66, 68, V2N2 1–2, 11, V2N4 101–2, 107, 110–11, 175; Philosophical V1N2 6, 23, 33–34, V1N3 5, 40–43, V1N4 15, 31–37, 40, 43–49, V1N6 19, V2N1 97, V2N2 1–13, V2N2 131–35, V2N6 3, 214

Deduction V1N2 5, 17, 33, 35–36, 40, V1N3 5, 15, 31, 33, 86, V1N4 28, 33, 44, 50, V2N2 3, 11–12, 49, V2N4 14, 30–31, 100; and Induction V1N2 14, 29, 33–36, 40, V1N3 15, 31–32, 36–37, 40–41, 47–48, V1N4 33–34, 50, V1N5 143, V2N2 13, V2N4 14, 31, 42, 52, 107; Limitative Theorems of V1N2 19–21, V1N5 80–81, 82–83, 91, 100, V2N3 81, V2N4 107–8; in Mathematics V1N2 2–3, 6, 13–14, 18, 22–23, 28–30, 41–42, V1N3 46–47, 50, 86, V1N5 79–81, V1N6 67–70, V2N2 1–2, 7, 11, V2N4 30, 94, 100, 102–3

Discovery; in Mathematics V1N2 3, 28–29, V1N3 49–50, V1N6 68–69, 112, V2N4 93–100; in Science V1N3 26, 30–32, 74–77, 89–90, V1N5 13–16, V2N2 121

Logic and Mathematics V1N2 2–6, 8–9, 15–17, 19–24, V1N3 46–47, 50, 76–77, V1N6 77, V2N4 106–8, 227

Mathematical Existence V1N2 3–9, V1N3 17, 46, 49–51, V1N4 11, V1N6 65, V2N4 99, 109–11

Mathematical Induction V1N2 41–42, V1N3 46–48, V1N5 72–73

Necessity; Logical V1N2 5, 8, 33–37, V1N3 5, 10–11, 15–16, 21–24, 27–28, 33–35, 40–41, 50, V1N4 8, 13, 15–17, 26, 43–52, 55–56, V1N5 112–13, V2N2 106, V2N4 227, 230, V2N5 184–85; Mathematical V1N2 2–6, 8–13, 37, 41–42, V1N3 17, 34–35, 46–47, 50–51, V1N6 56, V2N2 106–9, V2N4 96, 101–7; Metaphysical V1N2 33–35, V1N3 3, 5, 12, 15, 16–17, 21–27, 30–31, 40–43, V1N4 8, 16–17, 19, 26, 45, 49–52, 56–57, V1N5 112–13, V2N4 184–85, V2N5 98–100, 152, 155–56, 160–61, 230, V2N6 45–46, 78; Physical V1N2 35, V1N3 9–12, 21–40, V1N4 10–11, 15, 27, V1N5 72–73, V2N1 134, V2N2 107–9, V2N4 184–85, 190, V2N5 151–53, 159–61

Objectivity; and Logic V1N2 33–35, V1N3 40–41, V1N4 8, 19–23, 26–28, 59, V1N5 89–91, V1N6 98, 108, V2N1 134, V2N5 16, V2N6 208; in Mathematics V1N1 11–12, V1N2 3, 21–30, V1N3 49–51, V1N4 11, 28, V2N4 96, 98–99, 103, 108–11

Philosophy of Mathematics V1N2 1–30, V1N6 55–70, V2N4 99, 101–12; Conceptualism V1N1 11, V1N6 60, 62–64; Empiricism V1N2 13–15, 24–30, 42, V1N6 55–70, V2N4 104–7; Formalism V1N1 11; V1N2 18–19, 28; Intuitionism V1N1 11; V1N2 16–17, 28, V1N3 63–64, V1N6 60–61, V2N4 102–3, 107; Logicism V1N2 8–10, 15–16, 21, 28; Nominalism V1N1 10–11, V1N3 105, V2N4 109; Objectivism V1N1 11–12; V1N2 3, 21–30, V1N3 49–51, 104–6, V2N4 108–12; Platonism V1N1 11–12, V1N2 1–4, 24, V1N6 60–62, V2N4 108–10; Structuralism V1N2 4, V1N3 45, 50–51, V1N6 64–65

Possibility; Logical V1N2 8–9, V1N3 33–34, V1N4 26, 56, V2N1 1–2, 15–16, 28, V2N2 66–67, V2N4 227–28, 230, 232; Mathematical V1N2 6–7, 8–9, V1N3 17, 46, 49–51, V2N1 1–11, 13–14, 25, V2N4 227; Physical V1N3 9, 22–23, 25–27, 30, 33–40, V1N4 26–27, V2N1 1–3, 12–13, 20–28, V2N2 68–71, V2N4 99

Realism; Mathematical V1N1 5–6, 11–12, V1N2 1–7, 13–14, 24–30, 42, V1N3 49–51, V1N4 11, V1N6 60–62, V2N4 108–12; Metaphysical V1N2 1, 4, 22–24, V1N3 33–35, 44–45, V1N4 2–6, 38–39, V1N5 30–31, 35, 70, 89–90, V1N6 92, V2N4 28–31, 232, V2N5 16–17, V2N6 64; Scientific V1N3 6, 11–15, V1N4 61, V1N5 144–47, V1N6 49, 74–78, V2N4 2, 29–31, 46–47, 109

Relations; Logical V1N1 24, 30–31, V1N2 6, 9, 17, 21, 40, V1N3 34, 40–41, 44–46, V1N4 14–15, 20–23, 45, 63, V1N5 104, V1N6 77, V2N1 15–17, 134, V2N2 6–7, V2N3 81, V2N4 14, 82, 227–28, 106, V2N6 74–75, 90, 96–97, 106, 114, 184–85; Mathematical V1N1 6–7, 14–15, 33–34, V1N2 2–7, 9–10, 12–13, 29–30, 41, 98–100, 101–2, V1N3 17, 46–47, 101–10, V1N4 11, 16–17, 28, V1N6 61–63, 77, 178–79, V2N1 2–11, 21–26, 33–39, V2N2 1–2, 107–9, 119, 125, V2N3 79, 81, V2N4 103–7, 109–12, 170, 227, V2N6 133–34, 152–53, 172–74, 186; Physical V1N1 5, 11, 14–15, 28, 37–38, V1N2 12–13, 29–30, 98–100, V1N3 9–12, 28–30, 35, 39–40, 47–49, 71, 89, V1N4 15, 20–21, 66–78, V1N5 13–18, 71–76, V1N6 63, V2N1 27–28, 31–45, V2N2 26–27, 107–9, 113–26, V2N3 61–63, 68–70, 79–82, V2N4 102–7, 231, V2N5 18–21, V2N6 5, 131–86

Science; v. Logic V1N3 4, V1N4 33, 50–52, V2N6 184–85; v. Mathematics V1N2 28–30, V1N3 50, V1N6 77–78, V2N4 30, 104–7, 123, V2N5 9–10, 15; Mathematics in V1N1 5, 15, 27–28, V1N2 1, 30, V1N3 14, 39–40, 48–49, 104–5, 107–8, V1N5 13–17, V2N1 32–44, V2N2 27–28, 116–19, V2N3 52–54, V2N4 96, 105, 109, 123–27, 146, 149–57, 168, V2N5 9–10, 18–19, V2N6 131–86

Edited by Stephen Boydstun

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It seems this thread is a fairly good place to file this note. I want to let readers of Science and Mathematics section know that abstracts of the papers delivered last month at the biennial meeting of the International Society for the History of the Philosophy of Science have now been posted.

Specially Tantalizing in 1

Eric Audureau, "Remarks on the Relationship between Descartes’ Metaphysics and the Mathematical Concept of Gender of a Curve"

Kristian Camilleri, "Thought Experiments in Early Modern Science"

Mihnea Dobre, "Jacques Rohault and the Use of Experiment in Cartesian Physics"

Seffen Ducheyne, "Facing the Limits of Deductions from Phenomena: Newton’s Quest for a Mathematical-Demonstrative Optics"

Marco Giovanelli, "Indiscernibility. On Leibniz's Influence on Logical Empiricist Interpretation of General Relativity"

Guillaumin, "The distance to the Sun: Measurement as Fundamental Advance During the Scientific Revolution"

Specially Tantalizing in 2

James Lennox, "Aristotle on Norms of Inquiry"

Jane Maienschein, "Competing Embryological Epistemologies: How Do We Study Development?"

Craig Martin, ‘Forms and Qualities in Descartes’s Meteorological Explanations’

John McCaskey, "Bacon’s Idols and Harvey’s Eggs"

John McCaskey, "Whence the Uniformity Principle?"

Cornelis Menke, "John Stuart Mill on Predictions: The Whewell-Mill Debate"

Eric Schliesser, "The Weberian Roots of Chicago Economics"

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