In 1884, EA Abbot wrote a book titled Flatland: A Romance of Many Dimensions. It was about a world that existed only in two dimensions--only as a surface in which everyone lived (they did not live on it because that would imply a third, depth dimension). It was a satire on Victorian England's society, but it has often been cited in the context of our appreciation of dimensionality of existence.
We live in a 4-dimensional world: 3 space dimensions (which we can label x, y, and z, or north-south, east-west, and up-down), and one time dimension. As Einstein showed, all 4 are inextricably physical realities. We're quite familiar with this system, because we are of it, evolved in it, and live in it. These are often called 'orthogonal' dimensions, because you can move along one of them without affecting where you are on others; you can go east without affecting your north or up position (time is curiouser, but you can be in Cincinnati yesterday or tomorrow without affecting where Cincinnati is). Life is part and parcel of this, as well as inanimate energy and matter.
Mathematicians can deal with many more dimensions in abstract ways, and there are even some theoretical reasons to think there may be more physical dimensions that are simply out of our ken. At least, that's what 'string theory' asserts.
But we can think of 'dimensions' in other than ordinary physical ways. We can think of causation by different factors acting in different ways, each factor a separate measurement. If they are orthogonal (independent of each other) they hence can be treated as independent causes, data or statistical 'dimensions' with the same sort of mathematics used in physics and geometry. On the other hand it may be that theoretically, or empirically, multiple measured factors interact--are not independent.
Multivariate statistics is a common approach to such data, in which each independent risk factor, or independent set of correlated factors, is treated as a dimension, and every observed person has values in each dimension, the way you, now, have an east, north, up, and time position. You also have your age, gender, which are essentially independent, and your income and diet which are probably not. The same ideas apply to different parts of your genomes.
Multivariate statistical approaches categorize each observation's (say, each sampled person's) position and based on aggregate data, his/her possible future position. But as applied in real life, statistical estimates of these values are based on some relative criteria, comparing a person's position to the mean, or judging whether observations in this space are unusual in some way or other (statistically significance) that may be interpreted as indicating which dimensions are actually 'important'. Is this gene 'important' in heart disease? It's a subjective decision based on subjective evaluation of objective data.
Statistics is based on underlying probabilities, a subtle concept that can be interpreted in various ways. All science is based on statistical assessment in one way or another, because we never know all possible factors or observe anything with perfection. If we're lucky, the statistical aspects of things are about measurement error, not a poor understanding of reality.
Calculus is a mathematical way of expressing position, and especially changing position (movement) in any specified number of dimensions. Calculus and related mathematical tools are somewhat different from statistics because these are tools for studying the behavior of things whose laws of behavior we understand or at least specify. We make predictions or interpret observations based on a theory of how the world is. That theory is completely precise (even if it includes, as quantum mechanics does, fundamental probabilism). In a sense, a real scientific theory predicts data on the basis of some causative mechanism. That is what the 'law' is about.
By contrast, as we noted recently, statistical studies often do not have an underlying theory or law that is being tested or reflected in the data. The hope is, when the investigator even thinks about it explicitly, that observed correlations reveal something about the underlying causal processes that focused studies can reveal.
The Flatland issue comes about here because the mechanisms we can know to suggest must be confined to specific statements about the four dimensions of the physical world, or the essentially made-up 'dimensions' of statistical comparison, that we know about or can think of.
In statistical approaches, especially in areas of genomics, biomedicine and health, and evolution, we generally do not have an adequately precise theory. Regression analysis, for example, says that each increased unit of exposure to some risk-factor dimension increases risk by such-and-such an amount. But the dimensions are purely empirical, derived retrospectively from analyzing our collected data, that we may use to make predictions, but without those being based on a theory of mechanism. Often the included variables are arbitrarily chosen, and we have no way of knowing what might lie outside our chosen 'dimensions'--that our presence in Flatland prevents us from seeing. And again, what we include in much of these areas of science are generally not truly different dimensions, but instead are ways the data are arranged (for whatever reason) along the same dimensions. So if we measure something on dimension X and on dimension Y in each individual and find the data look like this...
....then we can (mathematically) make a 'new' dimension, the characteristics of the line shown, and say that the data mainly all are being measured on this one dimension. But it's really just a rearrangement of X and Y. The pattern may tell us something about the causal relationship between whatever X and Y are, and that may help us understand or simplify, but it doesn't get beyond the original dimensions. It is still commonsense, and a way of simplifying. It is a fundamental way of thinking in much of science and, relevant here, in genomics and evolutionary science.
But if this only gets us so far, is it because we are still viewing the world from Flatland, and seeing things only in terms of what, to us, is commonsense?
Are we living in Flatland?
These are generalizations that one may quibble about, but they generally characterize how business is done these days. So it is fair to ask whether there is reason to believe that, in any important ways, we are living in Flatland.
A few weeks ago, we enumerated 19 strange or paradoxical facts (well-known and not discovered by us!), that should give people pause, and ask whether we might, with our current statistical and other standard approaches, be living in a kind of Flatland, oblivious to dimensions that may be important but that we don't even know of--and whether we should at least be looking.
Does it matter?
For strong causal factors, Flatland is a fine place to be. We can identify those factors, and anticipate their effect with reasonable accuracy. Strong causal factors are typically relatively simple: they are always detectable, often act alone, and rarely fail to have their effect. These are the factors that, once identified, present juicy and suitable targets for engineering to avoid or modify them or their effects.
Humans are terrific engineers, so that when biomedical, agricultural, or other biologically related factors are at issue we can do something about them. If they are undesirable, we can detect and remove them. If they have desirable properties, we can use those properties to make changes in a positive direction (vaccines, bacteria that eat oil, enzymes and genes used to rearrange genomes experimentally or in public health). If they are genetic, we can use them to reconstruct history. Engineering may take time and skill, but it works eventually if we are determined enough.
When faced with such factors, there is no real detriment to living in Flatland. What we can't see, or don't even know is there to see, doesn't really matter.
But in much of biomedical and evolutionary genetics, there is reason to think that we are indeed living in Flatland, and that it does matter a whole lot. To see our world farther, we need to learn how to reach into its other causal dimensions.
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