Tag: mathematics

The Erdos Number

Paul Erdos, featured yesterday, chose a lifestyle that led to a striking number of shared work projects. Due to the sheer number of work friends, a number system was developed to keep track of the network that worked on shared ideas. Chat explains.

Paul Erdős, one of the most prolific mathematicians of the 20th century, collaborated with an extraordinary number of researchers throughout his life. His collaborators are often counted as part of the famous “Erdős Number” system, where Erdős himself has an Erdős Number of 0, his direct collaborators have a number of 1, their collaborators have a number of 2, and so on.

Estimated Number of Collaborators

Erdős collaborated with approximately 511 mathematicians on research papers during his lifetime. These collaborations resulted in over 1,500 papers, making him one of the most prolific authors in mathematical history.

This number of collaborators reflects Erdős’s unique approach to mathematics—he would travel extensively, visiting mathematicians worldwide, and work intensively with them on specific problems. This collaborative approach led to his reputation as a “mathematical nomad.”

Now, how do you think that work went when you think about all these math types puzzling over combinatorics or vertices of convex polygons? Did Erdos have a payroll and dole out cash? It seems it was the opposite. Collaborators and friends brought him into their home and put him up so he could work with them out of their university. This is not work compensated through pecuniary means.

So what’s in it for the collaborators? The Edos number, of course. Being in the Erdos network gives one sense of participation in the mathematical theory underway, and then their Erdos number specifies a claim to a distance from Erdos himself.

To recap, this type of work is voluntary and participatory, and the end product feeds into a jointly held asset—a school of thought in mathematics. Money is not the primary motivation for action. Membership in the network and the potential for the elevated position are the compensating factors. Every participant has access to the knowledge. It is a public good.

Here’s Chat’s visual.

Is it a public good to the whole world? In a sense, yes, but not in a practical sense. Just like it’s not practical to say the streets of Fargo, ND, are public to the whole world. The knowledge is open, but only a few will have the talents and learned knowledge to comprehend it. Only people in the geographic vicinity of Fargo will use their streets.

Is there externalizing and internalizing going on? Sure- when a new entrant learns a theorem, it becomes part of their knowledge. They have acquired the benefit, internalized, of the learned network. If a few of them collaborate on a textbook and sell it for their private pecuniary gain, they externalize knowledge and realize a gain. These actions do not conflict or reduce the network’s accomplishment. They add to the power and benefit of the group. The image you see inflates.

Paul Erdos’ life had living constraints, just as ours do. Yet the value of his research was such that he could be entertained at associates’ homes to assist in writing all 1,500 papers he left to the world.

Bayesian understanding and the Bible (speculative)

In probability theory and statistics, Bayes’ theorem (alternatively Bayes’ law or Bayes’ rule; recently Bayes–Price theorem[1]:44, 45, 46 and 67), named after the Reverend Thomas Bayes, describes the probability of an event, based on prior knowledge of conditions that might be related to the event.[2] For example, if the risk of developing health problems is known to increase with age, Bayes’ theorem allows the risk to an individual of a known age to be assessed more accurately (by conditioning it on their age) than simply assuming that the individual is typical of the population as a whole.

https://en.wikipedia.org/wiki/Bayes%27_theorem

What the Reverend Thomas Bayes came up with three hundred years ago has proven very useful as it allows predictions for future events based on a history of past events. Later on the concept was formalized by Pierre-Simon La Place as The Central Limit Theorem. Its statistical applications are used widely in medicine, pharmacology and finance.

But that’s not the part I find most interesting.

What we don’t know is Bayes’ philosophical aim in the hours he must have spent whittling away at untruths to reveal what endures. We have more insight into the thoughts of Richard Price, the man who shepherded Bayes’ manuscript to the Royal Society after his friend’s death.

What probably motivated Price to work on Bayes’ manuscript were the theological implications that Price perceived in the result. At this time in his life, Price was deeply immersed in theological and philosophical study. Price notes in Bayes (1763a) that Bayes had written an introduction to the paper; but Price did not include Bayes’ introduction and instead supplied his own. In other manuscripts of Bayes that I have seen (Bellhouse, 2002), Bayes typically gives no motivation for the mathematical results that he presents. The same may be true for his essay on probability. Price only says of Bayes that:


… his design at first in thinking on the
subject of it was, to find out a method
by which we might judge concerning the
probability that an event has to happen,
in given circumstances, upon supposition
that we know nothing concerning it but
that, under the same circumstances, it has
happened a certain number of times, and
failed a certain other number of times. He
adds, that he soon perceived that it would
not be difficult to do this… .

Later in the introduction to Bayes (1763a), Price states that:


Every judicious person will be sensible
that the problem now mentioned is by no
means merely a curious speculation in the
doctrine of chances, but necessary to be
solved in order to a sure foundation for all
our reasonings concerning past facts… .

Further on in the paper, after discussing de Moivre’s work, Price states:


The purpose I mean is, to shew what reason
we have for believing that there are in the
constitution of things fixt laws according to
which events happen, and that, therefore,
the frame of the world must be the effect of
wisdom and power of an intelligent cause;
and thus to confirm the argument taken from
final causes for the existence of the Deity.


What motivated Price to work on this paper was that to him the result provided a proof of the existence of God. Price came back to this theme in his theological work Four Dissertations (Price, 1767), which is mentioned by Morgan in the context of refuting Hume. A discussion of Price’s argument was given by Thomas (1977, pages 133 and 134).

The Reverend Thomas Bayes DR Bellhouse

Both Thomas Bayes and Richard Price were ministers and thus it is safe to assume they found truth in the teachings of Christianity. We might even speculate further that Bayes’ was trying his hand at a logical representation of the existence of the Holy Spirit among men; that Christians should trust one another to act in the ways of Jesus without the need for an immediate tally of deeds done.

For instance, consider a case of twelve neighbors living along a road; one might go to the effort of picking up the stray garbage; one neighbor may call the police at the sight of an intruder; one might petition the town to install a stop sign for safety. It really doesn’t matter which neighbor did which deed in consideration of the benefits to the street.

Such acknowledgements of individual skills, yet participation toward the work of a group, also appear in the Bible. Consider the passage in first Corinthians, Chapter 12 (King James Bible)

1 Now concerning spiritual gifts, brethren, I would not have you
ignorant.

2 Ye know that ye were Gentiles, carried away unto these dumb idols,
even as ye were led.

3 Wherefore I give you to understand, that no man speaking by the
Spirit of God calleth Jesus accursed: and that no man can say that
Jesus is the Lord, but by the Holy Ghost.

4 Now there are diversities of gifts, but the same Spirit.

5 And there are differences of administrations, but the same Lord.

6 And there are diversities of operations, but it is the same God which
worketh all in all.

7 But the manifestation of the Spirit is given to every man to profit
withal.

8 For to one is given by the Spirit the word of wisdom; to another the
word of knowledge by the same Spirit;

9 To another faith by the same Spirit; to another the gifts of healing
by the same Spirit;

10 To another the working of miracles; to another prophecy; to another
discerning of spirits; to another divers kinds of tongues; to another
the interpretation of tongues:

11 But all these worketh that one and the selfsame Spirit, dividing to
every man severally as he will.

12 For as the body is one, and hath many members, and all the
members of that one body, being many, are one body: so also is
Christ.

Don’t get caught up in the Christianity part, especially if you are not of the Christian faith. The understanding of the group is what is important. Everyone in the group is an individual with a variety of skills. When working on behalf of a joint mission, however, they become an indistinguishable member of the body. Like a drop of water in a river.

This is not the same as political socialism where a few at the top decide everyone is equal and push down resources in a fashion they deem equal. The starting point here is that everyone is blessed with differences, and can employ those differences and contribute them to a larger group. Once given, the reward is to each member equally. Just as each neighbor on the street benefits from living on a cleaner, safer street.

This is what I speculate Rev Bayes was trying to prove.