Article by: J J O'Connor and E F Robertson

Very little is known of the life of Zeno o

f Elea. We certainly know that he was a philosopher, and he is said to have been the son of Teleutagoras. The main source of our knowledge of Zeno comes from the dialogue Parmenides written by Plato.

Zeno was a pupil and friend of the philosopher Parmenides and studied with him in Elea. The Eleatic School, one of the leading pre-Socratic schools of Greek philosophy, had been founded by Parmenides in Elea in southern Italy. His philosophy of monism claimed that the many things which appear to exist are merely a single eternal reality which he called Being. His principle was that "all is one" and that change or non-Being are impossible. Certainly Zeno was greatly influenced by the arguments of Parmenides and Plato tells us that the two philosophers visited Athens together in around 450 BC.

Despite Plato's description of the visit of Zeno and Parmenides to Athens, it is far from universally accepted that the visit did indeed take place. However, Plato tells us that Socrates, who was then young, met Zeno and Parmenides on their visit to Athens and discussed philosophy with them. Given the best estimates of the dates of birth of these three philosophers, Socrates would be about 20, Zeno about 40, and Parmenides about 65 years of age at the time, so Plato's claim is certainly possible.

Zeno had already written a work on philosophy before his visit to Athens and Plato reports that Zeno's book meant that he had achieved a certain fame in Athens before his visit there. Unfortunately no work by Zeno has survived, but there is very little evidence to suggest that he wrote more than one book. The book Zeno wrote before his visit to Athens was his famous work which, according to Proclus, contained forty paradoxes concerning the continuum. Four of the paradoxes, which we shall discuss in detail below, were to have a profound influence on the development of mathematics.

Diogenes Laertius [10] gives further details of Zeno's life which are generally thought to be unreliable. Zeno returned to Elea after the visit to Athens and Diogenes Laertius claims that he met his death in a heroic attempt to remove a tyrant from the city of Elea. The stories of his heroic deeds and torture at the hands of the tyrant may well be pure inventions. Diogenes Laertius also writes about Zeno's cosmology and again there is no supporting evidence regarding this, but we shall give some indication below of the details.

Zeno's book of forty paradoxes was, according to Plato [8]:-

... a youthful effort, and it was stolen by someone, so that the author had no opportunity of considering whether to publish it or not. Its object was to defend the system of Parmenides by attacking the common conceptions of things.

Proclus also described the work and confirms that [1]:-

... Zeno elaborated forty different paradoxes following from the assumption of plurality and motion, all of them apparently based on the difficulties deriving from an analysis of the continuum.

In his arguments against the idea that the world contains more than one thing, Zeno derived his paradoxes from the assumption that if a magnitude can be divided then it can be divided infinitely often. Zeno also assumes that a thing which has no magnitude cannot exist. Simplicius, the last head of Plato's Academy in Athens, preserved many fragments of earlier authors including Parmenides and Zeno. Writing in the first half of the sixth century he explained Zeno's argument why something without magnitude could not exist [1]:-

For if it is added to something else, it will not make it bigger, and if it is subtracted, it will not make it smaller. But if it does not make a thing bigger when added to it nor smaller when subtracted from it, then it appears obvious that what was added or subtracted was nothing.

Although Zeno's argument is not totally convincing at least, as Makin writes in [25]:-

Zeno's challenge to simple pluralism is successful, in that he forces anti-Parmenideans to go beyond common sense.

The paradoxes that Zeno gave regarding motion are more perplexing. Aristotle, in his work Physics, gives four of Zeno's arguments, The Dichotomy, The Achilles, The Arrow, and The Stadium. For the dichotomy, Aristotle describes Zeno's argument (in Heath's translation [8]):-

There is no motion because that which is moved must arrive at the middle of its course before it arrives at the end.

In order the traverse a line segment it is necessary to reach its midpoint. To do this one must reach the ¹/₄ point, to do this one must reach the ¹/₈ point and so on ad infinitum. Hence motion can never begin. The argument here is not answered by the well known infinite sum

¹/₂ + ¹/₄ + ¹/₈ + ... = 1

On the one hand Zeno can argue that the sum ¹/₂ + ¹/₄ + ¹/₈ + ... never actually reaches 1, but more perplexing to the human mind is the attempts to sum ¹/₂ + ¹/₄ + ¹/₈ + ... backwards. Before traversing a unit distance we must get to the middle, but before getting to the middle we must get ¹/₄ of the way, but before we get ¹/₄ of the way we must reach ¹/₈ of the way etc. This argument makes us realise that we can never get started since we are trying to build up this infinite sum from the "wrong" end. Indeed this is a clever argument which still puzzles the human mind today.

Zeno bases both the dichotomy paradox and the attack on simple pluralism on the fact that once a thing is divisible, then it is infinitely divisible. One could counter his paradoxes by postulating an atomic theory in which matter was composed of many small indivisible elements. However other paradoxes given by Zeno cause problems precisely because in these cases he considers that seemingly continuous magnitudes are made up of indivisible elements. Such a paradox is 'The Arrow' and again we give Aristotle's description of Zeno's argument (in Heath's translation [8]):-

If, says Zeno, everything is either at rest or moving when it occupies a space equal to itself, while the object moved is in the instant, the moving arrow is unmoved.

The argument rests on the fact that if in an indivisible instant of time the arrow moved, then indeed this instant of time would be divisible (for example in a smaller 'instant' of time the arrow would have moved half the distance). Aristotle argues against the paradox by claiming:-

... for time is not composed of indivisible 'nows', no more than is any other magnitude.

However, this is considered by some to be irrelevant to Zeno's argument. Moreover to deny that 'now' exists as an instant which divides the past from the future seems also to go against intuition. Of course if the instant 'now' does not exist then the arrow never occupies any particular position and this does not seem right either. Again Zeno has presented a deep problem which, despite centuries of efforts to resolve it, still seems to lack a truly satisfactory solution. As Frankel writes in [20]:-

The human mind, when trying to give itself an accurate account of motion, finds itself confronted with two aspects of the phenomenon. Both are inevitable but at the same time they are mutually exclusive. Either we look at the continuous flow of motion; then it will be impossible for us to think of the object in any particular position. Or we think of the object as occupying any of the positions through which its course is leading it; and while fixing our thought on that particular position we cannot help fixing the object itself and putting it at rest for one short instant.

Vlastos (see [32]) points out that if we use the standard mathematical formula for velocity we have v = s/t, where s is the distance travelled and t is the time taken. If we look at the velocity at an instant we obtain v = ⁰/₀, which is meaningless. So it is fair to say that Zeno here is pointing out a mathematical difficulty which would not be tackled properly until limits and the differential calculus were studied and put on a proper footing.

As can be seen from the above discussion, Zeno's paradoxes are important in the development of the notion of infinitesimals. In fact some authors claim that Zeno directed his paradoxes against those who were introducing infinitesimals. Anaxagoras and the followers of Pythagoras, with their development of incommensurables, are also thought by some to be the targets of Zeno's arguments (see for example [10]). Certainly it appears unlikely that the reason given by Plato, namely to defend Parmenides' philosophical position, is the whole explanation of why Zeno wrote his famous work on paradoxes.

The most famous of Zeno's arguments is undoubtedly the Achilles. Heath's translation from Aristotle's Physics is:-

... the slower when running will never be overtaken by the quicker; for that which is pursuing must first reach the point from which that which is fleeing started, so that the slower must necessarily always be some distance ahead.

Most authors, starting with Aristotle, see this paradox to be essentially the same as the Dichotomy. For example Makin [25] writes:-

... as long as the Dichotomy can be resolved, the Achilles can be resolved. The resolutions will be parallel.

As with most statements about Zeno's paradoxes, there is not complete agreement about any particular position. For example Toth [29] disputes the similarity of the two paradoxes, claiming that Aristotle's remarks leave much to be desired and suggests that the two arguments have entirely different structures.

Both Plato and Aristotle did not fully appreciate the significance of Zeno's arguments. As Heath says [8]:-

Aristotle called them 'fallacies', without being able to refute them.

Russell certainly did not underrate Zeno's significance when he wrote in [13]:-

In this capricious world nothing is more capricious than posthumous fame. One of the most notable victims of posterity's lack of judgement is the Eleatic Zeno. Having invented four arguments all immeasurably subtle and profound, the grossness of subsequent philosophers pronounced him to be a mere ingenious juggler, and his arguments to be one and all sophisms. After two thousand years of continual refutation, these sophisms were reinstated, and made the foundation of a mathematical renaissance ....

Here Russell is thinking of the work of Cantor, Frege and himself on the infinite and particularly of Weierstrass on the calculus. In [2] the relation of the paradoxes to mathematics is also discussed, and the author comes to a conclusion similar to Frankel in the above quote:-

Although they have often been dismissed as logical nonsense, many attempts have also been made to dispose of them by means of mathematical theorems, such as the theory of convergent series or the theory of sets. In the end, however, the difficulties inherent in his arguments have always come back with a vengeance, for the human mind is so constructed that it can look at a continuum in two ways that are not quite reconcilable.

It is difficult to tell precisely what effect the paradoxes of Zeno had on the development of Greek mathematics. B L van der Waerden (see [31]) argues that the mathematical theories which were developed in the second half of the fifth century BC suggest that Zeno's work had little influence. Heath however seems to detect a greater influence [8]:-

Mathematicians, however, ... realising that Zeno's arguments were fatal to infinitesimals, saw that they could only avoid the difficulties connected with them by once and for all banishing the idea of the infinite, even the potentially infinite, altogether from their science; thenceforth, therefore, they made no use of magnitudes increasing or decreasing ad infinitum, but contented themselves with finite magnitudes that can be made as great or as small as we please.

We commented above that Diogenes Laertius in [10] describes a cosmology that he believes is due to Zeno. According to his description, Zeno proposed a universe consisting of several worlds, composed of "warm" and "cold, "dry" and "wet" but no void or empty space. Because this appears to have nothing in common with his paradoxes, it is usual to take the line that Diogenes Laertius is in error. However, there is some evidence that this type of belief was around in the fifth century BC, particularly associated with medical theory, and it could easily have been Zeno's version of a belief held by the Eleatic School.

Quote

Mysiq Genome Part 1


about the Mysiq Genome article, an introduction

ANALYSIS OF NUCLEAR RECEPTOR PSEUDOGENES IN VERTEBRATES:

(Adams et al. 2000), human (Robinson-Rechavi et al. 2001; Zhang et al. 2004), mouse (Zhang et al. 2004), and rat (Zhang et al. 2004). Pseudogenes (ψ) are nongenic DNA segments that exhibit a high degree of sequence similarity to functional genes but contain disruptive defects, including, not exhaustively, premature stop codons, splice site mutations, and frameshift mutations, that prevent them from being expressed properly. Disruption in the promoter regions of gene can also result in its pseudogenization. Based on whether they have gone through RNA processing, pseudogenes can be classified into two categories: Processed and unProcessed pseudogenes. Processed pseudogenes are generated by the integration of the reverse transcription products of Processed mRNA transcripts into the genome. UnProcessed pseudogene has not gone through RNA processing and thus has retained the original exon-intron structure of the functional gene. Previous studies have identified three NR pseudogenes in human: ψERRα (Sladek et al. 1997), ψHNF4γ (Tchenio, Segal-Bendirdjian, and Heidmann 1993), and ψFXRβ (Maglich et al. 2001; Otte et al. 2003) (See Table 1 for symbols and full names of NRs included in this study).

Recently several other NR pseudogenes were also identified in mice and rats (Zhang et al. 2004). However, the availability of eight vertebrate genome sequences (Waterston et al. 2002; Gibbs et al. 2004; International Chicken Genome Sequencing Consortium 2004; International Human Genome Sequencing Consortium 2004; Lindblad-Toh et al. 2005; The Chimpanzee Sequencing and Analysis Consortium 2005) makes it possible to conduct a detailed study of the NR pseudogenes in both human and vertebrate model systems. Here we present a comprehensive survey of NR pseudogenes in these eight vertebrate genomes and report their locations, sequences, and defects. Recently, pseudogenes in the entire human genome have been identified either in gene family-specific studies (Glusman et al. 2001; Zhang, Harrison, and Gerstein 2002) or in comprehensive surveys (Ohshima et al. 2003; Torrents et al. 2003; Zhang et al. 2003). Based on the mechanisms for pseudogene generation and the observations reported in those large-scale studies, we expected that NR pseudogenes would be mostly duplicated pseudogenes (like olfactory receptor pseudogenes) and few Processed ones as NR genes were created by multiple gene-duplication events and most NR genes have low expression levels. Our survey results here, however, are in striking opposition to these initial expectations. The analysis of these pseudogenes affords unique insights into the evolution and dynamics of this gene family and the mammalian genomes at large.

Results

Page 5

5  ||  similar, albeit defunct, copies of the transcript of the functional gene, which suggests a young age (38 Mya) for both of them.

As expected, we identified a set of NR pseudogenes in chimpanzee similar to those in human. However, the chimpanzee ortholog of the human ψERRα+ is absent. This absence indicates that ψERRα− was created first, at least before the divergence of human and chimpanzee, and at the same time the high sequence similarity and the shared defects between human ψERRα+ and ψERRα− suggest that the former was created by the duplication of the latter in the human lineage after its divergence from chimpanzee. In fact, those two pseudogenes reside in two expansive (>14.6-kb) and highly similar (96% identical) sequence segments in the human chromosome 13 that were created by a recent(< 6 million years ago), human-specific segmental duplication (Bailey et al. 2002; Cheng et al. 2005). Thus, human ψERRα+ is a duplication of a Processed pseudogene. This ‘duplicated-Processed’ pseudogene belongs to a new category of pseudogenes—first noted in a study of the human cytochrome c pseudogenes (Zhang and Gerstein 2003)—that are different from either duplicated or Processed pseudogenes in terms of their underlying generating processes. The original Processed pseudogene and the pseudogene duplicated from it both have little consequence to the fitness of the organism.

Nevertheless, they are distinct pseudogene species. The distinction made between them is important for estimating the frequency of retrotransposition of mRNA transcripts. Clearly, such estimation will be inflated if the 'duplicated Processed pseudogenes' are not excluded as they were generated by duplication, not retrotransposition, events. Human ψFXRβ is a unitary pseudogene with multiple nonfunctionalization mutations Previous studies (Maglich et al. 2001; Otte et al. 2003) have shown that human FXRβ is an unProcessed pseudogene with no functional counterpart (‘unitary pseudogene’) in the human genome. This gene was also nonfunctionalized in other Old World primates studied so far but encodes a functional receptor in other mammals (see (Otte et al. 2003) and below). The alignment of the mouse FXRβ protein sequence to the three-frame translation of the human genomic sequence reveals that the coding sequence of the original human FXRβ gene were interrupted by at least nine introns and in the currently defunct gene there are ten disruptive defects, which consist of three frame shifts, four nonsense mutations, and three splice site mutations (Figure 1). These defects are equally distributed at the beginning and the end of this pseudogene.

Human ψFXRβ and its mouse ortholog are located in two expansive (>25 Mb) syntenic regions in the two genomes (Figure 2). The same set of genes, in an identical order and orientation, in two genomic neighborhood make it unlikely that human FXRβ was inactivated by a chromosomal translocation or other genomic rearrangement processes. The comparison of the

Page 6

6  ||  orthologous sequences from human, chimpanzee, and rhesus (Figure 3A) reveals both ancestral and lineage specific sequence defects, 14 in all, in ψFXRβ from these three primates (Figure 3B).

The disruptive mutations at the first, second, and fourteenth positions in ψFXRβ are present in all three species, and hence most likely arose in the common ancestor of human, chimpanzee, and rhesus. Because the mutation at the fourteenth position, a nonsense mutation, is at the very end of the coding sequence and thus had considerably less disrupting power, either of the other two common mutations, one frame shift mutation and one splice site mutation at the start of the reading frame, could be the mutation that pseudogenized FXRβ in these primates. The orthologous genomic sequences from other primate species would make it possible to pin down the silencing mutation.

Based on four pairwise comparisons among the mouse and rat FXR and FXRβ sequences, our study dated the ancient gene duplication event that created this pair of paralogous genes to be 496 million years ago (Mya) prior to the speciation events (450 Mya) that ultimately gave rise to fishes and other vertebrates (Figure 4A). This estimation was confirmed by the search result for FXR and FXRβ in the genomes of representative species that both genes exist in human, chimpanzee, mouse, chicken, frog (Xenopus tropicalis), and fish (both zebrafish and pufferfish, Supplementary figure 1). The phylogeny of FXR and FXRβ reveals that by the measure of branch length (data not shown) FXRβ is evolving at least 5.6 times faster than FXRin mammals, but a similar difference in the evolution speed is not observed in non-mammalian vertebrates (Figure 4B, see Supplementary figure 2 for the multiple sequence alignment). Based on human, mouse, rat, and dog FXRβ sequences, our calculation indicates that the silencing of FXRβ happened 42 Mya.

Intergenic sequences immediately upstream and downstream to human ψFXRβ are conserved Human ψFXRβ is a transcribed pseudogene: real-time quantitative PCR detected relatively high levels of expression of its mRNA in testis (Maglich et al. 2001; Otte et al. 2003). This strongly suggests that the promoter and possibly other cis-acting elements that regulate the transcription of human ψFXRβ have remained largely intact and functional even long after the inactivation of ψFXRβ. Alignment of multiple genomic sequences from 14 vertebrates including human shows strong sequence conservation in the upstream noncoding regions— where regulatory elements may reside—of human ψFXRβ. Three highly conserved sequence segments, each 15 bp, were found within 250 bp immediately upstream to the ‘coding sequence’ of ψFXRβ (Figure 5A). Further upstream 4,500 bp away in an expansive (75 Kb) intergenic region between SIKE and SYCP1 resides a 250 bp sequence segment that is highly conserved across vertebrates between human and chicken (Figure 5B). This sequence segment has a high regulatory potential (>0.35, see (King et al. 2005)), and its mouse orthologous sequence is only 100 bp upstream to the first (noncoding) exon of the mouse FXRβ.