Tuesday, June 12, 2018

Bernard Bolzano


Bohemia (Now roughly the Czech Republic ) 1781-1848


Bernard Bolzano was many things, though principally a philosopher and mathematician. He was born in Bohemia, now the Czech Republic, and spoke and wrote German. He studied mathematics and philosophy at the Charles-Ferdinand University of Prague. Starting in 1800, he studied theology and graduated in 1804 (2). At this point, he had two positions open to him, as a professor of mathematics or a position in science and religion (2). Bolzano chose the science and religion route and was ordained in 1805, achieved his doctorate in philosophy and started working. 

However, he got into trouble because of his desire to reform the Austrian Empire. He thought war, and militarism in general, was terrible unnecessary and led a movement to refocus Austria on peace. (3, 4). At the time, most of Austrian society was built around the military and Bolzano wanted Austria's education and society to consider peace instead (3). He created a book on the subject, entitled On the Best State in which he imagined a government dedicated to reducing human suffering and misery as effectively as possible.  (1). 

As Napoleon was still attempting to conquer Europe at the time, Bolzano's views were taken none too well, and he was investigated for heresy. In 1819 he was drummed out of the university and banned from publication (3). He was eventually able to around the ban by getting some of his contacts and students to publish some of his work (1).
Bolzano achieved a level of insight into a variety of fields, mostly on a logical, mathematical, and philosophical basis. He tended towards existential issues, trying to establish concrete reasoning to get to the heart of the matter. Bolzano wrote in such diverse fields as the aesthetic, the theological, the logical, and the mathematical (1).

Bolzano wrote on the subject of art and beauty, and attempted to scientifically describe beauty. He was not actually successful. He argued that objects were beautiful if they "can give to all human beings properly developed in their cognitive faculties a pleasure, and the reason for this is that... [it] gives them at least an obscure intuition of the proficiency of their cognitive faculties" (1). As with some of his other philosophical work, it gives a definition, that leaves a lot to be desired. For instance, what does ‘proper development’ for cognitive faculties look like? Rather than being objective, his definition ends up being extremely subjective.

As for theology, he wanted a rational and credible lens through which to interpret miracles so he wrote extensively on probability theory in his theological works, trying to establish how to ensure the event had was improbable, that is less than 50% chance of occurring. He was not extremely concerned with the exact probability beyond that, just how to make sure it was sufficiently improbable to be considered a miracle (1). 

As for logic, Bolzano imagined three 'Worlds.' The first contains all material things, the second for psychological (and presumably emotional) ideas, and the third only for logical ideas and concepts. Then, he wanted to prove, without assumptions, that true statements existed (1). He created a proof to demonstrate there must exist an infinite  number of true statements existed. His proof flowed in a logical, almost mathematical way. 

Firstly, he examined the statement that there were no true statements, in and of themselves. Then, he examined the preposition that there are no true statements, found that must itself be a preposition and therefore proving at least one preposition, or statement not based on assumption exists. From there, he continued by plugging in the first result again. If there is one true statement, the statement that there is only one true statement means there is at least one MORE true statement and so on. Hence, an infinite number of true statements exist (1).

Among Bolzano's mathematical feats were creating a better proof for the bionomial theorem, applying probability to limits, and creating the Bolzano-Weierstrauss Principle (3, 5, 6). Of all his work, his mathematical contributions have been the most important. 


Works Referenced

  1. Bernard Bolzano https://plato.stanford.edu/entries/bolzano/
  2. Bernard Bolzano: Philosophy of Mathematical Knowledge http://www.iep.utm.edu/bol-math/
  3. Bernhard Bolzano https://www.britannica.com/biography/Bernhard-Bolzano
  4. Selected Writings on Ethics and Politics https://brill.com/view/title/30750
  5. The Philosophy of Bernard Bolzano: Logic and Ontology https://www.ontology.co/bolzanob.htm 
  6. The Bolzano-Weierstrauss Property and Compactness http://www.u.arizona.edu/~mwalker/econ519/Econ519LectureNotes/Bolzano-Weierstrass.pdf

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