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I was reading a book Laws and Symmetry by Bas C. Van Fraassen

I found that there is an argument for arguing that light travel in straight line:

Leibniz's reconstruction of these arguments goes roughly like this. Let it be given that the light travels from point A to point B; demonstrate that its path will be the straight line AB, if these points lie within an entirely homogeneous medium. This is the problem; how does one approach its solution? The problem itself introduces a geometric figure: the pair of points A, B, and the direction from A to B. To define any other direction in space, not along the line AB, one would need to refer to some other point, line, plane or figure, which has not been introduced in the given. Any rule governing the motion of light in this case must therefore either (a) imply that light follows the line AB, or (b) draw upon some feature X of the situation which could single out a direction not along that line. But the assumption that the medium is homogeneous, rules out the presence of such a feature X to be drawn upon. Therefore. . . .

We cannot quite yet conclude that therefore light travels along the straight line. As Leibniz clearly perceived, we need some bridge to get us from the fact that we could not possibly formulate any other rule here to the conclusion that light—a real phenomenon in nature—must follow that rule. The bridge, for Leibniz, is that the world was created by Someone who was in the position of having to solve these problems in the course of creation, and who would not choose without a reason for choice. If there was no such feature X to be preferred, obviously none of the choices of type (b) could then have been made. That God does not act without sufficient reason, implies that any correct science of nature, satisfies the constraint of Sufficient Reason. In the above problem, the conclusion that we cannot formulate any rule for the motion of light under these conditions, except that of rectilinear motion, yields then the corollary that light can follow no other path. The Principle of Sufficient Reason is introduced to fill the gap in the sort of arguments (the above, and also Hero's and Fermat's) here represented, and is in turn grounded in a certain conception of creation.

From my understanding, the basic idea behind is that when there is only two points $A,B$ defined in a homogeneous space (i.e. every point is the same). We can only draw a straight line between $A$ and $B$.

I was not a physics student, but I think of Euclid's postulates that given any two points, except drawing a straight line (Postulate 1), we can also draw a circle by using $AB$ as radius (Postulate 3).

Therefore, a light starting from point $A$ and given another point $B$, the light could in principle travel around $B$ at orbit (a circle).

Is this argument flawed? Can I conclude possible existence of Photon sphere without reference to any theory of gravity?

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Your own source says

Let it be given that the light travels from point A to point B.

The whole argument is prefaced with the restriction that we have light emitted from A and received at B. In your example, you want light to start at A and go around B, which is an entirely different thing altogether and has nothing to do with Leibniz.

Note also that Leibniz really only can claim to show that the photon path must be axisymmetric about the line AB. Only if take a photon to be perfectly localized in space and time does this then mean that it travels along the straight line in question.

Back to your claim of procognizing the photon sphere: The argument "X isn't obviously wrong, therefore X possibly exists" isn't really substantive, physically or philosophically. In fact, unless you have a definition of "possibly exists" that is somehow more strict than "isn't obviously wrong," this line of reasoning hasn't shown anything at all.

For a more extreme (but qualitatively identical) situation, imagine someone proposing "I assume the Pythagorean theorem, and I conceptualize the Higgs Boson, and I see no conflict between the two, so the Higgs Boson possibly exists." Sure the Higgs Boson possibly exists, and you knew that the moment you conceptualized it. Checking that the Pythagorean theorem doesn't disallow the Higgs Boson doesn't really accomplish much. Moreover, if you know nothing other than the Pythagorean theorem, it's questionable how much your concept of the Higgs Boson is what others take it to be.

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  • $\begingroup$ OK. I see the problem. In fact the argument isn't really conclude light travel in straight line. $\endgroup$
    – wh0
    May 6, 2015 at 2:28
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There's an argument I think briefly mentioned in Aristotles Physics; where he argues that an object travels in a straight line since to veer would require a cause; he actually says for what reason would it move up or down or to the left? In another sense it's an argument from symmetry; actually this is related to Newtons First Law where cause is interpreted as Force.

Similar reasoning were used by the Greek Atomists to argue for the Clinamen - this would be an inherent random motion of the atom; since they argued otherwise all atoms would move in a straight line and there would be no interactions.

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We can draw a circle using a line segment as a radius, but one of those pints that defines the line segments will be in the center. We cannot trace the circle and go from the center at the same time.

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  • $\begingroup$ we don't really need to draw the line segment, given any two points, except defining a line segment, we can define a circle. $\endgroup$
    – wh0
    May 6, 2015 at 2:19
  • $\begingroup$ @wonghang I mentioned it in that way because you stated that "I was not a physics student, but I think of Euclid's postulates that given any two points, except drawing a straight line (Postulate 1), we can also draw a circle by using AB as radius (Postulate 3). Therefore, a light starting from point A and given another point B, the light could in principle travel around B at orbit (a circle)." $\endgroup$
    – Jimmy360
    May 6, 2015 at 2:22
  • $\begingroup$ @wonghang so you are correct but what I said has to do with that, unless I'm misunderstanding what you are saying $\endgroup$
    – Jimmy360
    May 6, 2015 at 2:22

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