Today’s New Reason To Believe

This is Part 4 of a discussion on higher dimensionality. In Part 1 I mentioned a story by Robert A. Heinlein in which a four-dimensional cube (hypercube) played a role in the story. I promised there to provide some discussion of what was claimed to be true of hypercubes in the story. Well, here it begins.

Is there any simple way for the layperson to think about spatial dimensions higher than the three the mind is capable of? Can enough insight be gained to allow the average person to evaluate the merit of the “higher-dimensional” approach? The answer to these questions seems to be yes. However, the purpose of the short discussion that follows is not to be exhaustive, but simply to provide some examples in the realm of geometry that the reader can easily visualize. It should be noted that adding a new dimension to the treatment of a problem increases the complexity of possible solutions in ways far exceeding geometry. But, for anyone interested in examining the potential of this approach, geometry most easily provides an avenue for understanding a few of the added properties.

One of the unspoken axioms of mathematics holds that it is impossible for the human mind to visualize an object with dimensions higher than the three in which the mind resides. While it is possible, with mathematics, to analyze objects in higher dimensions, the mind cannot intuitively grasp their properties. So, for example, the author of the “house” story mentioned in Part 1 could use the properties of a four-dimensional cube to tell a good tale, but his architect could build only an “unfolded” cube. The only way to visualize objects in higher dimensions is to project (or alternately, unfold) those objects into a lower dimension.

I’ve drawn some figures (below) to help “see” the effects that come out of higher dimensions. In Figure 1 the first (leftmost) object is a single point. This represents a zero-dimensional (0-D) object because it has no length, width, or height (of course to draw it, as well as the other objects in the diagram, it is necessary to give it a little of each dimension.) The next object in the figure is a line, and is one-dimensional (1-D). It has length, but no other property. The third object is a square, and is obviously two-dimensional (2-D) in that it has both length and width. Finally, the fourth is a three-dimensional (3-D) cube that is projected into two dimensions. Sometimes a mathematician will refer to all four objects as “cubes” of different dimensions. So the square would be called a cube in two dimensions, while the line would be called a cube in one dimension. The point could be called a cube in zero dimensions, but laypeople may not find such a description meaningful. Perhaps that’s enough geometry for one day.

Figure 1: Objects in 0, 1, 2, and 3 dimensions

To be continued next week.

Latest Genetic Evidence Indicates No Interbreeding between Neanderthals and Humans

In the fall of 2006 geneticists from the University of Chicago created quite a stir when they announced evidence that a version of the human brain size gene, (microcephalin) came from Neanderthals as a result of interbreeding between these two species.

Microcephalin plays an important role in brain size development in humans. Mutations to this gene lead to microcephaly, a condition characterized by a severe reduction in brain volume. Evolutionary biologists think that changes in this gene through time led to the emergence of unusually large human brain size as hominids evolved into modern humans.

This prompted geneticists from the University of Chicago to examine the origin of this gene from an evolutionary perspective.

There are two categories of microcephalin genes, referred to as D and non-D haplotypes. The D haplotypes are found in over 70 percent of the world’s population, the non-D types in the remaining 30 percent. Based on their characterization, the researchers concluded that the D haplotypes originated about 37,000 years ago and that the non-D haplotypes emerged about 1.1 million years ago. They estimated that the two haplotypes diverged from one another about 1.7 million years ago.

They speculated that the non-D haplotypes uniquely emerged in the lineage that led to modern humans and the D haplotypes originated in the Neanderthal lineage. At 37,000 years ago, when Neanderthals and humans coexisted in Europe, interbreeding is thought to have introduced the D haplotypes into the human gene pool. The D haplotypes then rapidly spread among human people groups until it reached a frequency of about 70 percent.

Even though this evolutionary scenario seems compelling, it’s important to note that:

1. It is speculative
2. It represents indirect evidence for interbreeding, based on an inference

New findings reported at a recent Biology of Genomes meeting (Science, May 18, 2007, p. 967) stand in sharp contrast to the evolutionary scenario proposed by University of Chicago researchers. Through a direct analysis of the nuclear DNA sequences of the Neanderthal genome, researchers from Harvard University and the National Human Genome Research Institute conclusively discovered no indication whatsoever that humans and Neanderthals interbred. Analysis of human and Neanderthal Y-chromosome sequences by researchers from the Max Planck Institute for Evolutionary Anthropology in Leipzig, Germany, produced no evidence for gene flow between humans and Neanderthals. Earlier studies of Neanderthal mitochondrial DNA yielded a similar conclusion as well. Direct comparisons between Neanderthal DNA indicate that these two species did not interbreed.

This has important implications for studies like the one done by the University of Chicago geneticists.

Other researchers have interpreted unusual regions in the human genome with an apparently ancient date of origin as evidence for interbreeding between modern humans (who appeared on the scene recently) and archaic hominids. (For example, see here and here.) Again, it’s assumed these older genetic locations were introduced into the human gene pool as a result of interbreeding between humans and Neanderthals or humans and Homo erectus.

The latest results on human-Neanderthal interbreeding, however, invalidate these studies and strongly suggest that the methodologies and interpretative approaches used to conclude that humans and archaic hominids interbred must be flawed.

For a detailed discussion of how Neanderthals fit into RTB’s human origins model, see Who was Adam?

Last month I defined some terminology and provided a categorization of the different kinds of multiverses discussed by scientists. The least controversial multiverse, Level I, simply states that the universe does not end at the edge of the region observable by humans. The issue of the universe’s true size naturally arises.

Using the curvature of the universe measured by WMAP, scientists can put a lower bound, n, on the number of regions the size of our observable universe (referred to as Hubble volumes) that would fit in the universe. They estimate a minimum value of n=1000. However, if the current formulation of inflation proves correct, it predicts n=infinity for the Level I multiverse. That would seriously impact the strength of some apologetic arguments used for fine-tuning in the universe.

Consider the chances of drawing a royal flush from a standard deck of cards. Just four combinations of the 2,598,960 unique five-card possibilities make a royal flush. So the probability of drawing a royal flush on any single draw is 1 part in 649,740, but the odds grow with the number of draws available. If 10,000,000 draws occurred without a royal flush, one would wonder if the deck contained all the proper cards. An infinite number of draws assures a large number (actually an infinite number) of royal flushes.

Now consider the possible arrangements or states for the observable universe. Our Hubble volume (with a just-right Earth/Moon/Jupiter orbiting a just-right star at the just-right location in a just-right galaxy) corresponds to one state. Another state might look identical except Pluto does not exist. Many other states contain no planet remotely similar to Earth. One can envision a very large number of states in this manner, but the total number of possible states is finite. Consequently, the number of possible histories is also finite.

IF the universe is spatially infinite, Hubble volumes exhibiting all possible histories consistent with the laws of physics will exist somewhere—regardless of the improbability of any particular state! Humans exist only in the states meeting the requirements for their existence. The impact such a scenario has on many fine-tuning arguments is obvious. The question then becomes how does a Christian respond?

Here are a number of points to keep in mind:

1. An infinite Level I universe does not impact the fine-tuning arguments regarding the gross features of the universe such as the strengths of the four fundamental forces.

2. An infinite Level I universe does not argue against a Creator. It just removes one currently used apologetic argument.

3. An infinite Level I multiverse still relies on the current formulations of inflation but those formulations remain far from experimental verification. A comment by the cosmologist George Ellis regarding the multiverse applies here as well.

   The multiverse situation seems to fit St Paul’s description: “Faith is the substance of things hoped for, the evidence of things not seen.” In this case, it is faith that enormous extrapolations from tested physics are correct; hope that correct hints as to the way things really are have been identified from all the possibilities, and that the present marginal evidence to the contrary will go away.

4. Scientists often regard infinities arising in theories as a sign that the theory is breaking down. In this instance, many scientists embrace the spatial infinity predicted by inflation. Interestingly, the predicted actual spatial infinity derives from a reference-frame transformation involving a future potential time infinity.

The answer to whether the infinities of the Level I multiverse stand up under experimental scrutiny likely lies in the somewhat distant future. However, as Christians, we need to understand and feel the weight of these arguments in order to give an adequate response. As I will discuss next month, William Lane Craig argues that actual infinities do not exist because they lead to absurdities.