Category Archives: Logic

Mathematics and Logic

October 13, 2020 – 8:55 am

Large parts of this post are taken up with two subjects:

  1. The notion (due to Collingwood) of criteriological sciences, logic being one of them.

  2. Gödel’s theorems of completeness and incompleteness, as examples of results in the science of logic.

Like the most recent in the current spate of mathematics posts, the present one has arisen from material originally drafted for the first post in this series.

In that post, I defined mathematics as the science whose findings are proved by deduction. This definition does not say what mathematics is about. We can say however what logic is about: it is about mathematics quâ deduction, and more generally about reasoning as such. This makes logic a criteriological science, because logic seeks, examines, clarifies and limits the criteria whereby we can make deductions. As examples of this activity, Gödel’s theorems are, in a crude sense to be refined below, that

  • everything true in all possible mathematical worlds can be deduced;

  • some things true in the world of numbers can never be deduced;

  • the latter theorem is one of those things.

Continue reading →

By David Pierce | Also posted in Aristotle, Collingwood, Criteriological Science, Education, Euclid, Mathematics, Philosophy of Mathematics, Plato | Tagged , , , , , , , , , , , , | Comments (8)

Why It Works

September 26, 2020 – 10:49 am

The last post, “Knottedness,” constructed Alexander’s Horned Sphere and proved, or sketched the proof, that

  • the horned sphere itself is topologically a sphere, and in particular is simply connected, meaning

    • it’s path-connected: there’s a path from every point to every other point;

    • loops contract to points—are null-homotopic;

  • the space outside of the horned sphere is not simply connected.

This is paradoxical. You would think that if any loop sitting on the horned sphere can be drawn to a point, and any loop outside the horned sphere can be made to sit on the sphere and then drawn to a point, then we ought to be able to get the loop really close to the horned sphere, and let it contract it to a point, just the way it could, if it were actually on the horned sphere.

You would think that, but you would be wrong. Continue reading →

By David Pierce | Also posted in Exposition, Mathematics | Tagged , , | Comments (1)

Boolean Arithmetic

May 5, 2018 – 7:38 am

Mathematics can be highly abstract, even when it remains applicable to daily life. I want to show this with the mathematics behind logic puzzles, such as how to derive a conclusion using all of the following premisses:

  1. Babies are illogical.
  2. Nobody is despised who can manage a crocodile.
  3. Illogical persons are despised.

The example, from Terence Tao’s blog, is attributed to Lewis Carroll. By the first and third premisses, babies are despised; by the second premiss then, babies cannot manage crocodiles.

George Boole, The Laws of Thought (1854), Open Court, 1940

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By David Pierce | Also posted in Collingwood, Descartes, Exposition, G. H. Hardy, Language, Mathematics, Principles of Art | Tagged , , , , , , , , , , , , , , | Comments (7)

Hypomnesis

July 30, 2017 – 6:35 pm

When is a help a hindrance? The Muses have provoked this question. They did this through their agents, the cicadas, who sang around the European Cultural Center of Delphi, during the 11th Panhellenic Logic Symposium, July 12–5, 2017.

     Cicada, European Cultural Center of Delphi, 2017.07.15     

Cicada, European Cultural Center of Delphi, 2017.07.15

My question has two particular instances.

  1. At a mathematical conference, can theorems “speak for themselves,” or should their presenters be at pains to help the listener appreciate the results?

  2. When the conference is in Greece, even at one of the country’s greatest archeological sites, does this enhance the reading of ancient Greek texts, or is it only a distraction?

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By David Pierce | Also posted in Archeology, Archimedes, Art, Collingwood, Education, G. H. Hardy, Language, Mathematics, Philosophy, Plato, Principles of Art, Tourism, West Virginia | Tagged , , , , , , , , , , , | Comments (14)

NL VII: “Appetite”

May 5, 2014 – 3:18 pm

Index to this series

How can we compare two states of mind? This is the question of Chapter VII of The New Leviathan. The answer is contained in the chapter’s title. “Appetite” is a name, both for the chapter and for the fundamental instance of comparing a here-and-now feeling with a “there-and-then” feeling. We compare these two feelings because we are unsatisfied with the former, but prefer the latter.

It would seem then that appetite is at the root of memory. Thus we are among the ideas of the opening verses of The Waste Land of T. S. Eliot, who attended Collingwood’s lectures on Aristotle’s De Anima at Oxford (and was just a year older):

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By David Pierce | Also posted in Collingwood, Midgley, New Leviathan, Philosophy, Pirsig, T. S. Eliot, West Virginia | Tagged , , , , , | Comments (10)

NL V: “The Ambiguity of Feeling”

February 17, 2014 – 4:48 pm

Index to this series

Feeling differs from thought. Thought is founded in feeling; thought is erected on feeling; thought needs feeling. Thought needs feelings that are strong enough to support it. But thought itself is not strong (or weak); it has (or can have) other properties, like precision and definiteness. Thought can be remembered and shared in a way that feeling cannot.

The New Leviathan is a work of thought. One might say that a work of thought cannot properly explain feeling. Collingwood himself says this, more or less, in Chapter V, even in its very title: “The Ambiguity of Feeling.” Continue reading →

By David Pierce | Also posted in Aristotle, Collingwood, Descartes, Fowler, Grammar, History, Mathematics, New Leviathan, Philosophy, Philosophy of History, Plato, Principles of Art, Psychology, Science, T. S. Eliot | Tagged , , , , | Comments (7)

NL II: “The Relation Between Body and Mind”

January 20, 2014 – 11:39 am

Index to this series

I continue making notes on The New Leviathan of R. G. Collingwood (1889–1943). Now my main concern is with the second chapter, “The Relation Between Body and Mind”; but I shall range widely, as I did for the first chapter.

Preliminaries

Some writers begin with an outline, which they proceed to fill out with words. At least, they do this if they do what they are taught in school, according to Robert Pirsig:

He showed how the aspect of Quality called unity, the hanging-togetherness of a story, could be improved with a technique called an outline. The authority of an argument could be jacked up with a technique called footnotes, which gives authoritative reference. Outlines and footnotes are standard things taught in all freshman composition classes, but now as devices for improving Quality they had a purpose.

That is from Zen and the Art of Motorcycle Maintenance, chapter 17.

Does anybody strictly follow the textbook method of writing? Continue reading →

By David Pierce | Also posted in absolute presuppositions, Categorical Thinking, Collingwood, Descartes, Education, History, Mathematics, New Leviathan, Ontological Proof, overlap of classes, Philosophy, Philosophy of History, Pirsig, Plato, Science | Tagged , , , , , , , , , | Comments (13)

NL I: “Body and Mind”

January 13, 2014 – 2:35 pm

Index to this series. See also a later, shorter article on this chapter

The Chapter in Isolation

“Body and Mind” is the opening chapter of Collingwood’s New Leviathan. The chapter is a fine work of rhetoric that could stand on its own, though it invites further reading. In these respects it resembles the first of the ten traditional books of Plato’s Republic, or even the first of the thirteen books of Euclid’s Elements. The analogy with Euclid becomes a bit tighter when we consider that each chapter of The New Leviathan is divided into short paragraphs, which are numbered sequentially for ease of reference.

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By David Pierce | Also posted in absolute presuppositions, Collingwood, Criteriological Science, Descartes, dialectic, Education, Euclid, New Leviathan, Philosophy, Plato, Principles of Art, Psychology, Science | Tagged , , , , , , , , | Comments (18)

The Tradition of Western Philosophy

September 10, 2013 – 1:41 pm

Note added October 16, 2018: Here I compare two projects of re-examining the philosophical tradition named in my title. The projects are those of

  • R. G. Collingwood in An Essay on Philosophical Method (Oxford, 1933);

  • Stringfellow Barr and Scott Buchanan at St John’s College in Annapolis, Maryland, beginning in 1937.

I review

  • how I ended up as a student at St John’s;

  • how Collingwood has been read (or not read) by myself and others, notably Simon Blackburn;

  • how Collingwood’s Essay is based on the hypothesis of the “overlap of classes.”

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By David Pierce | Also posted in absolute presuppositions, Collingwood, Descartes, dialectic, Education, Euclid, History, Mathematics, overlap of classes, Philosophy, Philosophy of History, Pirsig, Plato, Principles of Art, Science, T. S. Eliot | Tagged , , , , , , , , , , , , , , | Comments (11)

Psychology

August 28, 2013 – 9:40 am

Preface (January 17–18, 2019). This essay is built around two extended quotations from Collingwood:

  1. From the posthumous Idea of History (1946) with the core idea, “people do not know what they are doing until they have done it.”
  2. From An Essay on Philosophical Method (1933), about how logic is neither a purely descriptive nor a purely normative science.

The quotations pertain to the title subject of psychology for the following reasons.

  1. Psychological experiments show that we may not know what we are doing until we have done it.
  2. Psychology is a descriptive science.

Psychological experiments can tell us about what we do, only when we presuppose the general applicability of their findings. This is true for any descriptive science. Philosophy demands more. A philosophical science like logic is categorical, in the sense of the second listed quotation, because it is what Collingwood will later call criteriological. I go on to discuss criteriological sciences as such in “A New Kind of Science,” but not here.

Continue reading →

By David Pierce | Also posted in Categorical Thinking, Collingwood, Criteriological Science, Euclid, Galileo, Ontological Proof, Philosophy, Principles of Art, Psychology, Science | Tagged , , , , , , | Comments (11)