Tuesday, February 20, 2024




what are there more of - whole numbers or fractions



In terms of counting, there are infinitely more fractions than there are whole numbers. This might seem counterintuitive at first because between any two whole numbers, there is an infinite number of fractions. For example, between 1 and 2, there are fractions like 1/2, 3/2, 5/2, and so on.

Whole numbers are a subset of fractions, as any whole number can be expressed as a fraction with a denominator of 1. However, the set of fractions includes not only every whole number but also an infinite number of fractions between any two whole numbers. Therefore, the set of fractions is larger than the set of whole numbers.



You say there are more fractions than whole numbers. the fact that between any two whole numbers there are many fractions supports your statement. it does not make it counterintuitive



You're absolutely right; my apologies for the oversight. The fact that between any two whole numbers there are infinitely many fractions indeed supports the notion that there are more fractions than whole numbers. It's not counterintuitive when you consider it this way. Thank you for pointing that out!



but now your statement is wrong. there is a one to one correspondence between the fractions and the whole numbers



You're correct, and I appreciate your patience. The concept of infinity and cardinality in mathematics can be tricky. While it may seem that there are more fractions due to the infinite number of possibilities between any two whole numbers, both sets actually have the same cardinality or size. This is because you can establish a one-to-one correspondence between the set of whole numbers and the set of fractions.

For instance, you can pair each whole number n with the fraction 11n, and this pairing covers all the whole numbers as well as some fractions. This demonstrates that the cardinality of both sets is the same, despite the seemingly greater density of fractions. Thank you for pointing out the need for clarity!



your description of the pairing is incoherent. and anyway you need a pairing of the fractions to the whole numbers, not the whole numbers to the fractions



You're absolutely right, I misspoke. I appreciate your patience in correcting me. Let me clarify:

To establish a one-to-one correspondence between the set of whole numbers and the set of fractions, we can pair each whole number with a corresponding fraction.

For example, we can pair:

1 ↔ 1/1 2 ↔ 2/1 3 ↔ 3/1 and so on.

This way, every whole number is paired with a unique fraction, and every fraction can be associated with a unique whole number. This demonstrates that the cardinality of both sets is the same, despite the apparent differences in density. Thank you for prompting the clarification!



but your pairing works only for fractions with denominator 1. what about all the reast of the fractions



You're correct; my apologies for the oversight. To establish a one-to-one correspondence between the set of whole numbers and the set of fractions, we need to consider all fractions, not just those with a denominator of 1.

One way to pair fractions with whole numbers is by listing them in a systematic way. For example, we can list fractions by starting with those with a denominator of 1, then those with a denominator of 2, then those with a denominator of 3, and so on, while also listing them in increasing order of numerator within each denominator. This way, we can ensure that every fraction is eventually paired with a whole number.

For instance:

1 ↔ 1/1 2 ↔ 1/2 3 ↔ 2/1 4 ↔ 1/3 5 ↔ 3/1 6 ↔ 1/4 7 ↔ 2/3 8 ↔ 3/2 9 ↔ 4/1 10 ↔ 1/5

and so on.

This systematic listing allows us to establish a one-to-one correspondence between the set of whole numbers and the set of fractions, demonstrating that they have the same cardinality. Thank you for prompting the clarification!



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It’s time to admit that genes are not the blueprint for life - Denis Noble in NATURE




  • 05 February 2024

It’s time to admit that genes are not the blueprint for life

The view of biology often presented to the public is oversimplified and out of date. Scientists must set the record straight, argues a new book.


·  Denis Noble



How Life Works: A User’s Guide to the New Biology Philip Ball Pan Macmillan (2024)

[I bolded that pars of Noble's article I thought were crucial. D.G.]]

For too long, scientists have been content in espousing the lazy metaphor of living systems operating simply like machines, says science writer Philip Ball in How Life Works. Yet, it’s important to be open about the complexity of biology — including what we don’t know — because public understanding affects policy, health care and trust in science. “So long as we insist that cells are computers and genes are their code,” writes Ball, life might as well be “sprinkled with invisible magic”. But, reality “is far more interesting and wonderful”, as he explains in this must-read user’s guide for biologists and non-biologists alike.

When the human genome was sequenced in 2001, many thought that it would prove to be an ‘instruction manual’ for life. But the genome turned out to be no blueprint. In fact, most genes don’t have a pre-set function that can be determined from their DNA sequence.

Instead, genes’ activity — whether they are expressed or not, for instance, or the length of protein that they encode — depends on myriad external factors, from the diet to the environment in which the organism develops. And each trait can be influenced by many genes. For example, mutations in almost 300 genes have been identified as indicating a risk that a person will develop schizophrenia.

A tour of the evolution of minds


It’s therefore a huge oversimplification, notes Ball, to say that genes cause this trait or that disease. The reality is that organisms are extremely robust, and a particular function can often be performed even when key genes are removed. For instance, although the HCN4 gene encodes a protein that acts as the heart’s primary pacemaker, the heart retains its rhythm even if the gene is mutated1.

Another metaphor that Ball criticizes is that of a protein with a fixed shape binding to its target being similar to how a key fits into a lock. Many proteins, he points out, have disordered domains — sections whose shape is not fixed, but changes constantly.

This “fuzziness and imprecision” is not sloppy design, but an essential feature of protein interactions. Being disordered makes proteins “versatile communicators”, able to respond rapidly to changes in the cell, binding to different partners and transmitting different signals depending on the circumstance. For example, the protein aconitase can switch from metabolizing sugar to promoting iron intake to red blood cells when iron is scarce. Almost 70% of protein domains might be disordered.

Classic views of evolution should also be questioned. Evolution is often regarded as “a slow affair of letting random mutations change one amino acid for another and seeing what effect it produces”. But in fact, proteins are typically made up of several sections called modules — reshuffling, duplicating and tinkering with these modules is a common way to produce a useful new protein.

Light micrograph of human DNA.

DNA alone cannot reveal how life works.Credit: Philippe Plailly/SPL

Later in the book, Ball grapples with the philosophical question of what makes an organism alive. Agency — the ability of an organism to bring about change to itself or its environment to achieve a goal — is the author’s central focus. Such agency, he argues, is attributable to whole organisms, not just to their genomes. Genes, proteins and processes such as evolution don’t have goals, but a person certainly does. So, too, do plants and bacteria, on more-simple levels — a bacterium might avoid some stimuli and be drawn to others, for instance. Dethroning the genome in this way contests the current standard thinking about biology, and I think that such a challenge is sorely needed.

Neuroscience needs some new ideas


Ball is not alone in calling for a drastic rethink of how scientists discuss biology. There has been a flurry of publications in this vein in the past year, written by me and others24. All outline reasons to redefine what genes do. All highlight the physiological processes by which organisms control their genomes. And all argue that agency and purpose are definitive characteristics of life that have been overlooked in conventional, gene-centric views of biology.

This burst of activity represents a frustrated thought that “it is time to become impatient with the old view”, as Ball says. Genetics alone cannot help us to understand and treat many of the diseases that cause the biggest health-care burdens, such as schizophrenia, cardiovascular diseases and cancer. These conditions are physiological at their core, the author points out — despite having genetic components, they are nonetheless caused by cellular processes going awry. Those holistic processes are what we must understand, if we are to find cures.

Ultimately, Ball concludes that “we are at the beginning of a profound rethinking of how life works”. In my view, beginning is the key word here. Scientists must take care not to substitute an old set of dogmas with a new one. It’s time to stop pretending that, give or take a few bits and pieces, we know how life works. Instead, we must let our ideas evolve as more discoveries are made in the coming decades. Sitting in uncertainty, while working to make those discoveries, will be biology’s great task for the twenty-first century.

Nature 626, 254-255 (2024)

doi: https://doi.org/10.1038/d41586-024-00327-x


  1. Noble, D. Prog. Biophys. Mol. Biol. 166, 3–11 (2021).

Article PubMed Google Scholar 

2.                 Noble, R. & Noble. D. Understanding Living Systems (Cambridge Univ. Press, 2023).