Abstract
Virtually all epistemologists agree that truth is a necessary condition for knowledge. Assuming that our three-dimensional world is nested within a higher-dimensional space, I use multidimensional geometry to present a type of case which undermines this fundamental principle of epistemology. I further argue that we would be unlikely to revise our epistemic practices in light of new discoveries even if our world turned out to be substantially different and many of our beliefs turned out to be false.
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Notes
See Gettier (1963).
One genuine exception is Popovic (2015), who directly argues that truth is not a necessary condition for knowledge and cites cases presumably belonging to the class of socially-constructed truths.
I am referring to spatial dimensions only. If one wishes to include a temporal dimension as well, the argument could be adjusted accordingly. For the sake of simplicity, I will build the initial example using the principles of Euclidean geometry (developed in Euclid’s Elements), but nothing of significance depends on it. We could have used any non-Euclidean approach (initially developed by mathematicians Janos Bolyai and Nikolai Lobachevsky). See Greenberg (1993) for a good summary of Euclidian versus non-Euclidean geometry. For a nice and entertaining discussion of this topic see Lewis Carroll’s (2009) Euclid and his Modern Rivals.
Of course, there is nothing specific about choosing four-dimensional space. We could have chosen five-dimensional or any other higher-dimensional space instead.
For an excellent treatment of those issues from the point of view of mathematics and physics see Rucker (2014). While it is certainly fair to say that we cannot possibly visualize a four-dimensional object, for an interesting attempt to provide guidance on how to conceive of higher-dimensional entities see McMullen (2014).
For an informal but wonderfully instructive explanation of this problem see Abbott’s (1992) Flatland.
This term was coined by a British mathematician C. H. Hinton (2015) in his A New Era of Thought, originally published in 1888. Tesseract is a four-dimensional analog of the cube. It is also sometimes referred to as tetracube, octachoron, 8-cell or cubic prism.
This is not the only possibility, of course. Face-first parallel projection of the tesseract into our three-dimensional space would result in a cuboid, edge-first parallel projection would give us a hexagonal prism, whereas vertex-first parallel projection would yield a rhombic dodecahedron. There are other possible outcomes as well, but we are assuming cell-first parallel projection that produces a cube.
We will also make the following assumption: If there are any forces acting on the intersection (the cube) coming from within our three-dimensional space, the tesseract moves accordingly so that the cube is subject to the same laws of physics as any other object in the three-dimensional world. Otherwise, the tesseract is stationary with respect to our three-dimensional space.
Presumably, it would require us to postulate that words refer to appearances. While this move might resolve the problem posed to the traditional analysis of knowledge by this particular case, it would inevitably limit our knowledge of the world to knowledge of the appearances. I doubt that many would be willing to accept this consequence. Of course, it is possible to work out a theory of reference that makes an exception only with respect to higher-dimensional objects. In principle, we could claim that in cases where we unintentionally try to pick out higher-dimensional objects, words refer to three-dimensional appearances, even though words otherwise refer to objects themselves. While there is some basis for making this exception due to our cognitive limitations, such a theory would almost certainly be criticized as being ad hoc and presented with obvious counterexamples. We could also attempt to fix this problem by reforming the language, but this is certainly not a promising direction for numerous reasons. Finally, we could try altering the concept of truth itself, but I sincerely doubt that this path would generate any serious interest.
Suppose that all higher-dimensional objects that we interact with are subject to the same specifications as made previously with respect to their relative movement. The purpose of this assumption is to prevent us from detecting any anomalies.
Certainly not all or even most of our beliefs need not turn out to be false. “The object in front of me is a cube” is false, but “I see a cube” or “this cube is made of metal” is presumably true. On the other hand, “the object in front of me is made of metal” need not be true if the properties of the object itself differ from the properties of its three-dimensional projection. There are also other numerous complications. For example, any sentence including the words “the biggest cube in the world” would likely end up being false, since it would in all likelihood pick out a cube not perceivable by as (an intersection of a different three-dimensional space and a higher-dimensional object, whereas this three-dimensional space is parallel to our three-dimensional world in an analogous way as two or more planes can be parallel within a three-dimensional universe). While it is difficult to estimate how many of our beliefs would end up being false, it is possible to amend the example and ensure that most of our beliefs are false if needed.
This type of discovery is genuinely possible. If an object “appeared” out of nowhere in the universe or suddenly started changing its shape or size, this would constitute evidence towards the hypothesis that higher-dimensional objects exist. A likely explanation of these observational phenomena would include a higher-dimensional object moving independently and intersecting with the three-dimensional space. Of course, this would require us to exempt at least one such object from the previous assumption.
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Popovic, N. The Tesseract, the Cube and Truthless Knowledge. Philosophia 48, 1569–1573 (2020). https://doi.org/10.1007/s11406-020-00172-y
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DOI: https://doi.org/10.1007/s11406-020-00172-y