# SHADOWS CLAIM PDF

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I find that I blithely claim self-evident status for lots of my own private opinions, my interpretations of fact, and a motley assortment of other sentiments. It goes without saying that I do all this without any justification; I have no need of it!

Even with that, special care must be taken in defining what it means for points to be inside or outside the closed loop. On a piece of paper meant to stand for the Euclidean plane draw, with one stroke of the pen, a circle, or any closed loop.

Now try to connect a point inside the drawn circle to a point outside the circle by some continuous curve. Now the exercise is to think about the possible status of this simple construction and the claim following it. To pin them down, we must put them into some kind of formal context.

This notion gets to an essential difference between mathematics and other realms of thought. Among all the other things that it is, mathematics is the art of the unambiguous. It is almost uncanny how mathematics has the capability of achieving such a high level of exactitude in its definitions and assertions. Often much is made in mathematical logical circles of how crucial it is for theories to be consistent.

This is true enough, but it is a piece — admittedly, the essential piece — of a larger day-to-day issue that arises when thinking about mathematics; namely, the importance of being sensitive to ambiguities of all sorts and not only to the crisis that would occur if the same proposition were to be provably both true and false.

Here is an easy mathematical example that might give you a sense of how finely tuned mathematicians are to the question of ambiguity. Compare that situation with a pair of triangles GHI and JKL that are congruent, where one of the triangles hence the other as well is isosceles.

## The Allegory of the Cave

Now there is a structural difference between these two situations; modern algebra strongly presses home to its practitioners how important it often is to keep such differences in mind. Namely, in the first, scalene, case the congruence itself between ABC and DEF is unique in the sense that there is only one way of pairing the vertices of the first with the vertices of the second so as to achieve a perfect overlap of the two triangles.

To rephrase our example a bit more generally and succinctly: It can be important to distinguish between knowing that two objects are merely equivalent, and knowing more firmly that they are equivalent via a uniqueequivalence. I imagine that the legal profession has lots to teach scientists and mathematicians about the subtly different levels of plausibility in inferential arguments.

Mathematicians may be known for the proofs they end up discovering, but they spend much of their time living with mistakes, misconceptions, analogies, inferences, partial patterns that hint at more substantial ones, rules of thumb, and somewhat systematic heuristics that allow them to do their work. How can one assess the value of any of this mid-process?

Here are three important modes of plausible reasoning. I recommend that for fun you take a look at a few pages of this classical expository treatise on plausible inference in mathematics:. Princeton University Press, — This may have seemed harmless until Bertrand Russell showed that such kinds of insouciant universal quantification led to contradictions. Harvard University Press, For very a different and interesting!

## Kresley-Cole-Shadow's-Claim-Excerpt

Bloomsbury, If a line segment intersects two straight lines forming two interior angles on the same side that sum to less than two right angles, then the two lines, if extended indefinitely, meet on that side on which the angles sum to less than two right angles. Specifically, see:.

He went on to say:. The models are then tested, and experiments confirm or falsify theoretical models of how the world works.

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This is the way science has worked for hundreds of years. Scientists are trained to recognize that correlation is not causation, that no conclusions should be drawn simply on the basis of correlation between X and Y it could just be a coincidence.

Instead, you must understand the underlying mechanisms that connect the two. Once you have a model, you can connect the data sets with confidence. Data without a model is just noise.

But faced with massive data, this approach to science hypothesize, then model, then test is becoming obsolete. Consider physics: Newtonian models were crude approximations of the truth wrong at the atomic level, but still useful. A hundred years ago, statistically based quantum mechanics offered a better picture but quantum mechanics is yet another model, and as such it, too, is flawed, no doubt a caricature of a more complex underlying reality.

Now biology is heading in the same direction. There is now a better way.

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Petabytes allow us to say: We can analyze the data without hypotheses about what it might show. We can throw the numbers into the biggest computing clusters the world has ever seen and let statistical algorithms find patterns where science cannot.

But correlation alone will never replace the explanatory power of mathematics. No wonder, then, that to teach mathematics is so formidable a task; you are grappling with the pure essence of learning. To teach effectively, you had better use every species of true persuasion, of genuine evidence, that you can bring to the picture, with the rigor of proof as its frame.

Barry Mazur is a mathematician at Harvard University currently working in number theory.

For discussions of similar issues, but with attention paid to changes in attitudes towards foundations, see:. Home Articles. Evidence in Mathematics The Plan People in mathematics, or in the sciences related to mathematics, know well that the issue of evidence, as it pertains to how one studies math, or engages in its practice, or does research in it, is not a simple matter.

And unless you plotted them very accurately, they would show up as blobs in the plane with nothing particularly interesting about their perimeters — something like this: Here is a little thought exercise in mathematical framing: Well Defined This notion gets to an essential difference between mathematics and other realms of thought.

A heuristic method is one that helps us to actually come up with possibly true, and interesting statements and gives us reasons to think that they are plausible. A non-heuristic method is one that may be of great use in shoring up our sense that a statement is plausible once we have the statement in mind, but is not particularly good at discovering such statements for us.

I recommend that for fun you take a look at a few pages of this classical expository treatise on plausible inference in mathematics: Specifically, see: For a preliminary discussion of the fifth postulate. For a rundown of attempts to prove it. He went on to say: See also: Semi-Rigorous Mathematics? Borwein, J. Borwein, R. It is striking, though, how different branches of knowledge — the humanities, the sciences, mathematics — justify their findings so very differently; they have, one might say, quite incommensurate rules of evidence.

Often a shift of emphasis, or framing, of one of these disciplines goes along with, or derives from, a change of these rules, or of the repertoire of sources of evidence, for justifying claims and findings in that field. Law has, of course, its own precise rules explicitly formulated.

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These types of judgments frame the project of evolution. If we accept that the shape and mood of a field of inquiry is largely, or even just somewhat, determined by the specific kinds of evidence needed to have consensually agreed-upon findings or results in that field, it becomes important to study the perhaps peculiar nature of evidence in different domains to appreciate how these distinct domains fit into the greater constellation of intellectual effort.

It was structured as an extended conversation between different practitioners and our students. A number of experts contributed to lectures and discussions. We found it very useful to learn in some specificity from people in different fields, via concrete examples graspable by people outside the field, what evidence consists of in physics, economics, biology, art history, history of science, mathematics and law.

Once one looks with a microscope at the structure of evidence in any of these fields, even though this structure is quite specific to the field, and a moving target, the project of understanding it in a larger context is very much worth doing.

Evidence in Mathematics The Plan People in mathematics, or in the sciences related to mathematics, know well that the issue of evidence, as it pertains to how one studies math, or engages in its practice, or does research in it, is not a simple matter.

I also believe that it is just a good idea to become aware of the various distinct profiles of evidence in a number of disciplines, and to understand how these profiles change in time, how they shape and delimit and define those disciplines, and how they influence the interaction between disciplines.

Such reflection may enable any of us to appreciate and view our own work in a deeper way. That theory studied the structure of certain regions in the plane that are very important for issues related to dynamics. Surely, Fatou or Julia would not have been able to make too exact a numerical plot of these regions. And unless you plotted them very accurately, they would show up as blobs in the plane with nothing particularly interesting about their perimeters — something like this: Partly due to the ravages of World War I, and partly because of the general consensus that the problems in this field were essentially understood, there was a lull of half a century in the study of such planar regions, now often called Julia sets.

From such pictures alone it became evident that there is an immense amount of structure to the regions drawn and to their perimeters. This almost immediately re-energized and broadened the field of research, making it clear that very little of the basic structure inherent in these sets had been perceived, let alone understood.

It also suggested new applications. Computers nowadays, as we all know, can accumulate and manipulate massive data sets. But they also play the role of microscope for pure mathematics, allowing for a type of extreme visual acuity that is, itself, a powerful kind of evidence. In the early s, the mathematician John McKay made a simple observation. What is peculiar about this formula is that the left-hand side of the equation, i. The relevant issue, for us, is given by the following general ground rules of the game: you are faced with three closed doors in a row, onstage and are told that the goat is behind one of those doors, there being nothing behind the other two.

All you have to do is open the right door, i. But you are asked, first, to indicate what door you intend to open, without actually opening it.

Are your chances of winning independent of whether or not you change your preliminary choice? Or is there a clear preference for one of the two strategies: sticking to your guns, or switching?

Here is an exercise: Come up with an answer to the question above and give an argument defending it. The point for us is that evidence in mathematical reasoning sometimes arises in slant ways. The structure of this simple game reminds me of a certain computational strategy that makes judicious choices from an inventory of different algorithms, and switches from one algorithm to another depending on new clues that come up in mid-computation.

But these are only three of the many forms in which evidence presents itself in pure mathematical research. Of course, the physical sciences have been providing evidence for mathematical truths beginning with the earliest astronomical observations to the most current work in string theory.

A tricky notion, self-evidence: I find that I blithely claim self-evident status for lots of my own private opinions, my interpretations of fact, and a motley assortment of other sentiments.

It goes without saying that I do all this without any justification; I have no need of it! Even with that, special care must be taken in defining what it means for points to be inside or outside the closed loop.

Construction: On a piece of paper meant to stand for the Euclidean plane draw, with one stroke of the pen, a circle, or any closed loop.

Now try to connect a point inside the drawn circle to a point outside the circle by some continuous curve. Claim: Your continuous curve will, at some point, intersect the circle. Now the exercise is to think about the possible status of this simple construction and the claim following it.

To pin them down, we must put them into some kind of formal context. Well Defined This notion gets to an essential difference between mathematics and other realms of thought.

Among all the other things that it is, mathematics is the art of the unambiguous.

## Kresley-Cole-Shadow's-Claim-Excerpt

It is almost uncanny how mathematics has the capability of achieving such a high level of exactitude in its definitions and assertions. Often much is made in mathematical logical circles of how crucial it is for theories to be consistent.To rephrase our example a bit more generally and succinctly: It can be important to distinguish between knowing that two objects are merely equivalent, and knowing more firmly that they are equivalent via a uniqueequivalence.

But faced with massive data, this approach to science hypothesize, then model, then test is becoming obsolete. Sign In. We found it very useful to learn in some specificity from people in different fields, via concrete examples graspable by people outside the field, what evidence consists of in physics, economics, biology, art history, history of science, mathematics and law.

For a shadow is nothing at all, only the reflection of a man—strewn across the ground. The woman was just Poetry. We can throw the numbers into the biggest computing clusters the world has ever seen and let statistical algorithms find patterns where science cannot. De- tractors, contrarily, have criticized the canon for its elitism and chauvin- ism, or for its claimed purity, nonpoliticality, and aestheticism.

Thanks to Leo Treitler for introducing me to this picture. Why is the sky blue?

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