# Dimensions

## 19 posts in this topic

I'm not educated in modern physics, so I wonder what is meant when I hear references to all kinds of different dimensions that different theories posit. The only dimensions that I know of are the three spatial dimensions. Those I have no problem with. I've heard of time referred to as a fourth dimension, but I'm not sure that's a valid application of the word dimension. I have no idea what anything beyond that would even mean.

So how many dimensions are there? Is it proper to refer to time as a dimension, or this this just a metaphorical use? What is the definition of dimension? What do the people who accept other dimensions actually believe in? Are dimensions sometimes treated as alternate realities (like in sci-fi) or are they something else? Thanks for the clarification.

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I'm not educated in modern physics, so I wonder what is meant when I hear references to all kinds of different dimensions that different theories posit. The only dimensions that I know of are the three spatial dimensions. Those I have no problem with. I've heard of time referred to as a fourth dimension, but I'm not sure that's a valid application of the word dimension. I have no idea what anything beyond that would even mean.

So how many dimensions are there? Is it proper to refer to time as a dimension, or this this just a metaphorical use? What is the definition of dimension? What do the people who accept other dimensions actually believe in? Are dimensions sometimes treated as alternate realities (like in sci-fi) or are they something else? Thanks for the clarification.

It should be noted that the word dimension has a usage in mathematics that needn't imply any direct physical interpretation. To be very informal the dimension of an abstract mathematical space is the number of "degrees of freedom" it has. For example the Hilbert space specifying the set of possible states of a quantum system is infinite dimensional but these dimensions don't have the same physical interpretation that the three dimensions of our everyday space do. I'm not an expert on Objectivism but from what others have said I believe this usage of the word dimension is what most Objectivists would call a "concept of method" or an "epistemological device".

As far as physical dimensions go, according to the theory of special relativity time is indeed a physical dimension along with the three spatial dimensions. One way of thinking about this is as follows. One of the key facts about special relativity is that the theory is invariant under the Lorentz group - this is the transformation corresponding to a change in speed. If T is a time coordinate and X is a spatial coordinate then a Lorentz boost in the direction of the X axis takes the form:

T' = (cosh r)T + (sinh r)X

X' = (sinh r)T + (cosh r)X

where r is some parameter. Physically this is the transformation between two reference frames that have relative motion along the X-axis. (I've suppressed the other two spatial dimensions Y,Z for simplicity.)

The key thing to take away from these transformations is that it mixes up the spatial and temporal coordinate. In order to know the new coordinate T' you have to know both T and also X. And similarly to know the new coordinate X' you have to know not only X but also T.

Or to put it another way: the Lorentz group is only a symmetry on spacetime (space and time considered together) not individually on space (X) or time (T).

Explaining it in different terms: how would you answer someone who claimed that left/right and back/forth were dimensions but up/down wasn't a dimension and thus the world is 2-dimensional? Well one answer would be that to exclude up/down is arbitrary because after all what one considers up/down is dependent on your orientation. If you choose an up/down axis then rotate your system then the notion of up/down changes. There is no rotation invariant way of choosing up/down so to exclude one direction from being a dimension is arbitrary, given that the laws of physics are rotation invariant. Well exactly the same argument holds for time in the context of special relativity. If you tried to consider only X,Y,Z as dimensions and exclude time T then doing so would require an arbitrary choice for T, which would change after you Lorentz boosted the system.

(One should contrast this with Galilean relativity, a theory that preceeded special relativity. In this theory transforming between frames in relative motion is governed by Galilean transformations not Lorentz transformations and it turns out it _is_ possible to choose a time axis that is Galilean invariant. So it made sense to separate time from space in this context.)

Hence T is properly considered a dimension along with X,Y,Z in the context of SR, essentially because the laws of SR are Lorentz invariant.

The "extra dimensions" in the context say of String Theory are highly speculative. I'm not personally completely dismissive of such theories, because of a number of technical reasons, but one can't go wrong by remaining very skeptical of such possibilities at the present state of knowledge.

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I'm not educated in modern physics, so I wonder what is meant when I hear references to all kinds of different dimensions that different theories posit. The only dimensions that I know of are the three spatial dimensions. Those I have no problem with. I've heard of time referred to as a fourth dimension, but I'm not sure that's a valid application of the word dimension. I have no idea what anything beyond that would even mean.

So how many dimensions are there?

That depends on the context of your question. The physical world of our perception is 3-dimensional, but, mathematically, there is no limit to the number of dimensions that one can devise. Here is a response to a similar question that I gave earlier, in this post

OK, 3+ dimensions are methodological, not metaphysical.

1. What does, eg, 7-dimensional manifold, do for math? Please explain as simply as, eg, infinity is the ability to always count one more.

To begin with, in simple terms a mathematical manifold is like a space, a little universe we create mathematically. A space has a certain number of dimensions. In grade school you made plots or graphs using an x-y coordinate system. That would be a 2-dimensional space, and a 2-dimensional manifold is the abstract version of what you drew on paper. I'm sure you have seen pictures of objects that are referenced to an x-y-z coordinate system, perhaps a three-dimensional view of a familiar object that is rotated around. A simple 3-dimensional manifold would be the abstraction consisting of an x-y-z space. Although most of us can only directly visualize a 3-dimensional space, there is no reason that, mathematically, we cannot extend that space to 4 dimensions, 5 dimensions, or more.

Leaving aside the many esoteric mathematical uses of a multidimensional manifold, there are no end of practical applications of this. For instance, a mathematical economist may create a space consisting of a large number of variables, each variable a separate dimension in the manifold. The economist is creating a multidimensional model so see the behavior of functions in this world of many variables, with the eventual goal of relating the results to real-world behavior.

Another practical application is one I developed, growing out of the need to analyze the results of actual experiments. For instance, say we prepare a biological specimen capable of surviving for several days in vitro, and we want to study the movement of cells as they emanate from the neural tube of an embryo. Using scanning laser microscopy we can take image slices on a set time period, building up a digital library of 3-dimensional data over time. This time-dependent data can then be thought of as a 4-dimensional manifold, and all sorts of analysis, from image registration to cell calculations can be performed, all in this 4-dimensional world.

2. Then time is not part of our three-dimensional world (height, width, depth) because its not a property of entities but relational?

Yes, time is a relational concept, not a metaphysical existent. But that fact does not obviate using time as an added dimension for, say, the normal 4-dimensional manifold of special and general relativity. This is the usual concept of spacetime, which is a perfectly valid mathematical abstraction We operate in the abstract spacetime manifold as a model, and wind up predicting real-world behavior in the real physical universe within which we live.

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As far as physical dimensions go, according to the theory of special relativity time is indeed a physical dimension along with the three spatial dimensions.

I completely disagree that in special relativity time is a "physical dimension." But, I have explained and argued against that notion so many times before, here on THE FORUM and elsewhere, that all I have the energy for is to note my disagreement.

T' = (cosh r)T + (sinh r)X

X' = (sinh r)T + (cosh r)X

where r is some parameter. Physically this is the transformation between two reference frames that have relative motion along the X-axis.

No. It is mathematically a transformation, not "physically" a transformation. Mathematics has implications for the physical world, but it is not the same as the physical.

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OK, so why are there exactly 3 physical, spatial dimensions? I.e. what are the implications of there being 3 dimensions?

(For example, if there were only one dimension, for any entity to move, it would have to pass through its neighboring entity -- i.e. it would have to simultaneously occupy the same space.)

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OK, so why are there exactly 3 physical, spatial dimensions? I.e. what are the implications of there being 3 dimensions?

(For example, if there were only one dimension, for any entity to move, it would have to pass through its neighboring entity -- i.e. it would have to simultaneously occupy the same space.)

Are you questioning why there is specifically 3 dimensions, or are you questioning why there is any specific number of dimensions?

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OK, so why are there exactly 3 physical, spatial dimensions? I.e. what are the implications of there being 3 dimensions?

(For example, if there were only one dimension, for any entity to move, it would have to pass through its neighboring entity -- i.e. it would have to simultaneously occupy the same space.)

Spacial dimensions are not things, they are relationships between entities. You start with the fact that there are entities and observe the relations between them based on what is. We observe that there are three independent measurements required to completely specify a spacial relationship between points and develop a mathematics of geometry based on that. Asking "why is reality what it is?" or "why can't it be something else?" are not valid metaphysical questions. You can only ask "why" to mean what is the causal relation based on the nature of what reality is.

The idea of a "two dimensional space" is a mathematical abstraction in which you ignore the thickness as irrelevant for certain purposes. There is no such thing as an entity with an actual zero thickness; it would not exist.

"Dimension" in mathematics has a completely different technical meaning than the three spacial dimensions, namely, the number of independent quantities in some aggregate specification. A vector in mathematics has some number of independent components and need not refer to physical space at all. An n dimensional "space" in mathematics is a technical abstration in which the n components refer to n measurements of something, which may or may not refer to the spacial dimensions of physical reality, along with technical rules for how to combine the vectors in various kinds of computations. In mathematics there are even "infinite dimensional spaces", defined in terms of appropriate limiting processes for sequences of numbers used to specify something else. Such n dimensional and infinite mathematical "spaces" are abstractions of method and have nothing to do with metaphysical speculations superseding our observed reality of entities located with 3 measurements of geometry. It is a different, technical, use of the term "space" as a concept of method which applies to our ordinarily observed three dimensional space as a special case.

The use of vectors with 4 components in relativity means that the vectors account for 4 independent measurements: time and all three spacial dimensions. It does not mean that time is a 4th physical dimension physically equivalent to dimensions in space. Time is a measure of change; the "points" at different times do not simultaneously exist the way different points in space do. But the mathematics of vectors is a legitimate method for relating space and time in terms of a single abstract concept, the 4-vector. (The method is further complicated by the fact that it requires using complex numbers to represent time multiplied by "i" = sqrt(-1) and the speed of light for dimensional and physical consistency, so time does not mathematically appear in the same form as space -- this is entirely a matter of mathematical method relating quantities.)

You can visualize a path in two of the three spacial dimensions as a function of time in a 3-D diagram showing a curve in 3 dimensions, but that is a mathematical representation of spacial position as a function of time; you can't reify the abstration and conclude that your graph means time is a "place" like spacial position in reality.

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The idea of a "two dimensional space" is a mathematical abstraction in which you ignore the thickness as irrelevant for certain purposes. There is no such thing as an entity with an actual zero thickness; it would not exist.

An entity with zero thickness would not exist in the 3-dimensional world of our perception. However, you cannot in advance specify such a requirement for the existence of the ultimate constituents of reality.

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An entity with zero thickness would not exist in the 3-dimensional world of our perception. However, you cannot in advance specify such a requirement for the existence of the ultimate constituents of reality.

What evidence is there for the possibility of such a thing? I wouldn't "rule it out", but neither would I speculate about it in any way at all. Our whole concept of 2-D space is based on an abstraction of our concept of space in the world of our perception. Without evidence for something else, I would not speculate about what dimensions could or could not be ruled out metaphysically because to think in such terms at all would be to use stolen concepts. If some physicists have some knowlege based on particle physics that at least gives a rational tentative approach to try as a hypothesis, that's a different matter. We can't constrain science to the already known, but neither is the arbitrary permissible.

I don't even know what "space" would mean at that level of abstraction about the super-microscopic. Maybe it wouldn't apply at all to the extent of such things in the normal sense of relations between entities, except for some abstractly formulated notion about some sense of position of the thing itself, without regard to its own extent in one or more dimensions. Whatever will eventually be known about such things will be highly abstract, highly mathematical and inferred, not directly observable the way we observe entities with three dimensions, and it would be a mistake to speculate based only on our observational knowledge regarding space. But that isn't the same thing as saying we can't "rule out" something for which two (or any other number) of independent quantities specifiy something about it in some more advanced idea than we have now. Leaving it at that level of generality is the same as saying we can't rule what we don't know, but neither should we arbitrarily speculate one way or the other.

The important point in the context of Nate's question is to firmly keep the distinction between the three dimensions of our world of perception of entities and the mathematical methods of n-dimensional "spaces", to maintain the hierarchy of abstractions, and to not reify the mathematical sense of dimension into alleged evidence of n != 3 dimensions of physical entities.

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An entity with zero thickness would not exist in the 3-dimensional world of our perception. However, you cannot in advance specify such a requirement for the existence of the ultimate constituents of reality.

What evidence is there for the possibility of such a thing?

It is not I who asserted that such a thing does or does not exist. I was simply correcting your assertion that "There is no such thing as an entity with an actual zero thickness; it would not exist," as not applying to the ultimate constituents.

I wouldn't "rule it out" ...

But that is exactly what you did when you said "it would not exist."

Look, I realize that you probably meant your remark to apply in the 3-dimensional world of our perceptions. But, for the sake of clarity and understanding, I wanted to make clear that when it comes to the ultimate nature of reality, to its ultimate constituents, we cannot place upon them the requirement you made and claim "it would not exist."

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So how many dimensions are there? Is it proper to refer to time as a dimension, or this this just a metaphorical use? What is the definition of dimension? What do the people who accept other dimensions actually believe in? Are dimensions sometimes treated as alternate realities (like in sci-fi) or are they something else? Thanks for the clarification.

If the manifold in which you are working is a vector space, the the dimension of the manifold is the cardinality of the smallest set of vectors which spans the space. That is any vector in the space can be expressed as a linear combination of vectors in the spanning set.

For example a space in which each point consists of three co-ordinates of location and three co-ordinates of momentum is a six dimensional phase space for the point so described. If a system consists of N points then the phase space has dimension 6*N. That is to say each points position and velocity is given.

As to the four dimensional space of events you have three co-ordinates for location (where) and one co-ordinate for time (when), hence a four dimensional space.

In a database suppose you have 20 fields for the index that locates a piece of data in the database uniquely. Then that data set is a 20 dimensional "space". It's not geometric but it uses the same principle of specifying a point as is used in analytic geometry.

Bob Kolker

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So how many dimensions are there? Is it proper to refer to time as a dimension, or this this just a metaphorical use? What is the definition of dimension? What do the people who accept other dimensions actually believe in? Are dimensions sometimes treated as alternate realities (like in sci-fi) or are they something else? Thanks for the clarification.

If the manifold in which you are working is a vector space, the the dimension of the manifold is the cardinality of the smallest set of vectors which spans the space...

Unfortunately that does not address Nate's question, the formulation of which clearly showed that he does not have the knowlege of advanced physics and mathematics to plunge into something that begins "If the manifold in which you are working..." in the name of explanation. Nor does it answer the question he asked at all. His question has already been answered in understandable terms, although he did not say if it was sufficient for his purposes or if further elaboration is required. Further discussion is always welcome, but tossiing out out-of-context abstract technical terms presented as floating abstractions is not helping. Such a rationalistic approach is typically the cause of the confusion in the first place, and raises the question of whether Bob himself understood the nature and source of the question and the kind of answer required.

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So how many dimensions are there? Is it proper to refer to time as a dimension, or this this just a metaphorical use? What is the definition of dimension? What do the people who accept other dimensions actually believe in? Are dimensions sometimes treated as alternate realities (like in sci-fi) or are they something else? Thanks for the clarification.

If the manifold in which you are working is a vector space, the the dimension of the manifold is the cardinality of the smallest set of vectors which spans the space...

Unfortunately that does not address Nate's question, the formulation of which clearly showed that he does not have the knowlege of advanced physics and mathematics to plunge into something that begins "If the manifold in which you are working..." in the name of explanation. Nor does it answer the question he asked at all. His question has already been answered in understandable terms, although he did not say if it was sufficient for his purposes or if further elaboration is required. Further discussion is always welcome, but tossiing out out-of-context abstract technical terms presented as floating abstractions is not helping. Such a rationalistic approach is typically the cause of the confusion in the first place, and raises the question of whether Bob himself understood the nature and source of the question and the kind of answer required.

Speaking as moderator -- and as someone who, like most FORUM members, is not a professional mathematician or physicist -- let's cut out the jargon.

Common sense and the level of knowledge an intelligent layman would have is the assumed context around here. Anything else hinders communication.

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mathematician or physicist -- let's cut out the jargon.

Common sense and the level of knowledge an intelligent layman would have is the assumed context around here. Anything else hinders communication.

One person's jargon is another person's precise and correct definition. The notion of dimension (in the sense the original poster intended) is a highly technical matter and cannot be addressed fully using everyday locutions. From the time of Euclid to the time of Solomon Lefschetz who did define dimension generally and correctly nearly 2300 years of common sense reigned and got some of the mathematics and physics wrong. Mathematics was rescued from its logical imprecision* only in the last 200 years and physics did not get a hold of the world of the very large AND the very small until maybe the time of Maxwell when field theory and atomic theory were beginning to "gel". In every instance "common sense" took a severe beating. (Even Isaac Newton assaulted "common sense" by proposing action at a distance). Einstein said something to the effect that common sense is the set of prejudices and preconceived notions we acquire prior to the age of eighteen. Einstein himself was the arch enemy of common sense, as his Special Theory of Relativity clearly indicates. He did a "number" on time and space.

In a general and not too technical way, dimension is the number of independent parameter values that must be given to specify an object in the domain being discussed. In the case of space, three parameters are required to specify instantaneous position. In the space of events four parameters are required to identify an event, three for where and one for when. Etc. Etc. So in that sense, time (as read off a fiduciary clock) IS a dimension. To describe the kinetics of a single particle no less than six parameters are required, three of position, and three of momentum. In this case both position and momentum are required and momentum is a function of time and place.

Please forgive this discourse into mathematics and physics but if we are going to discuss the matter correctly (even if not thoroughly) we do have to be precise. And precision -does- require a specialized technical vocabulary. I will now take off my teacher's cap and gown and return to podium to "assumed context". Thank you for your time and attention.

Bob Kolker

* For example Euclid's Elements is shot through with logical imprecision which was not corrected until Hilbert produced his seminal -Grunlagen der Geometrie- in 1899.

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mathematician or physicist -- let's cut out the jargon.

Common sense and the level of knowledge an intelligent layman would have is the assumed context around here. Anything else hinders communication.

One person's jargon is another person's precise and correct definition.

I'm all for precision, which is why one should not use a term that the target audience is not familiar with, give examples the audience cannot follow because they do not know the referents, or cite authorities the audience does not have the knowledge to evaluate as trustworthy or not.

Observe that my late husband Stephen would present his ideas on physics one way on a technical forum like TEWLIP (where the audience was primarily other physicists), much less technically here on THE FORUM, and even less abstractly on his "Mr. Science" spots on a morning radio show. It comes down to recognizing and respecting that different audiences have different contexts of knowledge and tailoring one's presentation accordingly.

The notion of dimension (in the sense the original poster intended) is a highly technical matter and cannot be addressed fully using everyday locutions.

That may very well be the case. If so, that is a very good reason why such a discussion may not be appropriate for THE FORUM -- at least on a highly technical level.

Perhaps a simple explanation followed by a pointer to more in-depth presentations would be the way to go. Again, see Stephen's presentations of scientific topics here on THE FORUM for examples of how this can be done.

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This is not just a matter of 'jargon', but of relevance and objectivity. Explanation requires integrating the new with the old, already known. You don't plunge into abstract statements and undefined technical terminology without regard to the base knowledge presumed by the question. If Nate had the knowledge required to understand Bob's statements and convert them into an answer he would have asked a different question. To toss out technical terms with an expectation of providing a 'self-contained' argument without regard to the facts that give rise to the concepts used and the appropriate context of knowledge is to deal in floating abstractions and rationalism, not 'precision'.

As I wrote previously, "Such a rationalistic approach is typically the cause of the confusion in the first place, and raises the question of whether Bob himself understood the nature and source of the question and the kind of answer required." That is not answered by a claim to superior "precision" allegedly requiring the jargon.

There was nothing imprecise about the discussion already given in answer to Nate's question. 'Precision' is not philosophical rationalism, including its form as free-floating mathematics. Rationalism, no matter how 'precise', is still rationalism. It should not be 'package dealed' in the name of 'precision', as it so often is, especially in presentations of mathematics. Invoking such names as Hilbert and Einstein to disparage 'comon sense' does not help and is not relevant.

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There was nothing imprecise about the discussion already given in answer to Nate's question. 'Precision' is not philosophical rationalism, including its form as free-floating mathematics. Rationalism, no matter how 'precise', is still rationalism. It should not be 'package dealed' in the name of 'precision', as it so often is, especially in presentations of mathematics. Invoking such names as Hilbert and Einstein to disparage 'comon sense' does not help and is not relevant.

It is very relevant. Every major advance in physics has beaten common sense to a bloody pulp.

Bob Kolker

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There was nothing imprecise about the discussion already given in answer to Nate's question. 'Precision' is not philosophical rationalism, including its form as free-floating mathematics. Rationalism, no matter how 'precise', is still rationalism. It should not be 'package dealed' in the name of 'precision', as it so often is, especially in presentations of mathematics. Invoking such names as Hilbert and Einstein to disparage 'common sense' does not help and is not relevant.

It is very relevant. Every major advance in physics has beaten common sense to a bloody pulp.

If by "common sense" you mean the axioms of identity and causality, then it certainly has not. Advances in physics have been made by those who respect the axioms -- the rules of reality -- despite the crazy "explanations" offered by those who don't.

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This is not just a matter of 'jargon', but of relevance and objectivity. Explanation requires integrating the new with the old, already known. You don't plunge into abstract statements and undefined technical terminology without regard to the base knowledge presumed by the question. If Nate had the knowledge required to understand Bob's statements and convert them into an answer he would have asked a different question. To toss out technical terms with an expectation of providing a 'self-contained' argument without regard to the facts that give rise to the concepts used and the appropriate context of knowledge is to deal in floating abstractions and rationalism, not 'precision'.

As I wrote previously, "Such a rationalistic approach is typically the cause of the confusion in the first place, and raises the question of whether Bob himself understood the nature and source of the question and the kind of answer required." That is not answered by a claim to superior "precision" allegedly requiring the jargon.

There was nothing imprecise about the discussion already given in answer to Nate's question. 'Precision' is not philosophical rationalism, including its form as free-floating mathematics. Rationalism, no matter how 'precise', is still rationalism. It should not be 'package dealed' in the name of 'precision', as it so often is, especially in presentations of mathematics. Invoking such names as Hilbert and Einstein to disparage 'comon sense' does not help and is not relevant.

It is very relevant. Every major advance in physics has beaten common sense to a bloody pulp.

Which does not address what I said. You do not represent a 'major advance in physics', and neither Hilbert nor Einstein are relevant to the issue here: Ayn Rand's philosophy is not merely 'common sense' either, but the common sense originally appealed to by Betsy was the simple ability to address basic questions without extraneous jargon.

Whether or not the results of any particular scientific breakthrough -- even if correctly formulated -- is surprising, there are a lot more ways to go wrong than to get something right at all levels of knowledge. A statement that sounds bizarre is no sign that it is right or appropriate. In this thread, Nate asked the simple question of the meaning of dimension, which was answered months ago in terms appropriate to the knowledge assumed by the question. As described previously, one does not in the name of explanation throw out technical examples with no meaning to the questioner, plunging in with "If the manifold in which you are working...". If and when he acquires the appropriate technical knowledge to understand more advanced topics he can absorb the corresponding technical meaning of 'dimension' for those cases. Knowledge is hierarchical, not floating abstractions manipulated rationalistically in the name of 'precision'. Invoking the names of Einstein and Hilbert do not excuse rationalism ignoring what is required for explanation. This is not a matter of Bob Kolker, lonely misunderstood scientific genius bravely battling, in the name of Einstein and Hilbert, the ignorant masses on the Forum rejecting unfamiliar new ideas. Rhetoric about 'beating common sense to a bloody pulp' may be relevant, but not in the way you think.