## February 28, 2008

### Peirce on Mathematics

#### Posted by David Corfield

By the winter of 1897-8, the jobless philosopher Charles Peirce was financially crippled, continuing his studies despite the cold, unable to heat his house. Regretting this situation, William James, at Harvard, organised for Peirce to give a series of lectures for much needed remuneration. Peirce wanted to talk about formal logic, but James persuaded him that this would severely reduce his audience (“Now be a good boy and think a more popular plan out. I don’t want the audience to dwindle to 3 or 4…”), and that he would be better off discussing “topics of a vitally important character”.

He did adjust the content somewhat, as you can see in the published lectures Reasoning and the Logic of Things, (Kenneth Ketner ed., Harvard University Press, 1992), where you can read some of the correspondence between Peirce and James.

In the first lecture, Peirce outlines his understanding of the architectonic structure of the sciences, dividing them into the psychical and the physical. He notes that on both of these sides, the sciences are growing increasing nomological (law-like), tending towards the state of general psychics and general physics. These in turn are “surely developing into parts of metaphysics,” and

Metaphysics in its turn is gradually and surely taking on the character of a logic. And finally logic seems destined to become more and more converted to mathematics.

Naturally, Peirce then asks “And now whither is mathematics tending?” He notes that “Mathematics is based wholly upon hypotheses, which would seem to be entirely arbritrary.” And continues by observing that nobody now can cover the whole field, and that most important developments are made by specialists. Then,

For that reason you would expect the arbritrary hypotheses of the different mathematicians to shoot out in every direction into the boundless void of arbritrariness. But you do not find any such thing. On the contrary, what you find is that men working in fields as remote from one another as the African Fields are from the Klondike, reproduce the same forms of novel hypotheses. Riemann had apparently never heard of his contemporary Listing. The latter was a naturalistic Geometer, occupied with the shapes of leaves and birds’ nests, while the former was working upon analytical functions. And yet that which seems the most arbitrary in the ideas created by the two men, are one and the same form. This phenomenon is not an isolated one; it characterizes the mathematics of our times, as is, indeed, well-known. All this crowd of creators of forms for which the real world affords no parallel, each man arbitrarily following his own sweet will, are, as we now begin to discern, gradually uncovering one great Cosmos of Forms, a world of potential being. The pure mathematician himself feels that this is so. He is not indeed in the habit of publishing any of his sentiments nor even his generalizations. The fashion in mathematics is to print nothing but demonstrations, and the reader is left to divine the workings of the man’s mind from the sequence of those demonstrations. But if you enjoy the good fortune of talking with a number of mathematicians of a high order, you will find that the typical Pure Mathematician is a sort of Platonist. Only, he is Platonist who corrects the Heraclitan error that the Eternal is not Continuous. The Eternal is for him a world, a cosmos, in which the universe of actual existence is nothing but an arbitrary locus. The end that Pure Mathematics is pursuing is to discover that real potential world. (pp. 120-1)

This allows him a dig at James – “Once you become inflated with that idea vital importance seems to be a very low kind of importance, indeed.”

Posted at February 28, 2008 2:28 PM UTC

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### Re: Peirce on Mathematics

A tragic life indeed. Apparently Peirce had enemies in high places, who blocked his ability to win the academic appointments he so richly deserved.

Peirce also suffered from a facial neuralgia which is believed to have been trigeminal neuralgia, one of the most painful medical conditions known. (When triggered by, e.g., a kiss to the cheek or a light breeze, an attack is typically likened to a violent electric shock lasting several seconds – and there may be dozens or even hundreds of such episodes in the course of a day. Before effective treatments became available, it was known as the “suicide disease”, for evident reasons.) In the Wikipedia article on Peirce, it is averred that this condition may have substantially contributed to his growing isolation in his later years.

Posted by: Todd Trimble on February 28, 2008 3:24 PM | Permalink | Reply to this

### Re: Peirce on Mathematics

The line taken in the editorial to the book of lectures I mentioned on the source of this enmity is disapproval of his relationship with his future second wife Juliette before his divorce.

I see the Wikipedia entry refers to his biographer’s conjecture “about various psychological and other difficulties”. He seems to have be taught by his father a little like John Stuart Mill was.

Benjamin [Peirce] took a special personal interest in his son’s education, emphasizing independent, self-directed study and sometimes keeping him up all night in exercises to develop his concentration. (Reasoning and the Logic of Things, 4)

Posted by: David Corfield on February 29, 2008 9:38 AM | Permalink | Reply to this

### Jealous student blocks career; Re: Peirce on Mathematics

Peirce and his family, and a jealous student who rose to power, limited his career cruelly. The key paragraph form the wikipedia entry on Simon Newcomb [12 March 1835 - 11 July 1909] whose birthday it is today as I blog this is:

“Newcomb studied mathematics under Benjamin Peirce and the impecunious Newcomb was often a welcome guest at the Peirce home[2] However, he later became envious of Peirce’s talented son, Charles Sanders Peirce and has been accused of a “successful destruction” of C. S. Peirce’s career.[3] In particular, Daniel Coit Gilman, president of Johns Hopkins University, is alleged to have been on the point of awarding tenure to C. S. Peirce, before Newcomb intervened behind the scenes to dissuade him.[4] About twenty years later, Newcomb allegedly influenced the Carnegie Institution Trustees, to prevent C.S. Peirce’s last chance to publish his life’s work, through a denial of a Carnegie grant to Peirce, even though Andrew Carnegie himself, Theodore Roosevelt, William James and others, wrote to support it.[5]”

# Brent, J. (1993). Charles Sanders Peirce: A Life. Bloomington: Indiana University Press. ISBN 0253312671.
# ^ Brent (1993) p.288
# ^ Brent (1993) p.128
# ^ Brent (1993) pp150-153
# ^ Brent (1993) pp287-289

Posted by: Jonathan Vos Post on March 12, 2008 3:11 PM | Permalink | Reply to this

### Physics and “threeness”; Re: Peirce on Mathematics

One of my favorite paragraphs by Pierce, which may connect with this blog’s musings on duality and triality in foundational Mathematical Physics:

“Ancient mechanics recognized forces as causes which produced motions as their immediate effects, looking no further than the essentially dual relation of cause and effect. That was why it could make no progress with dynamics. The work of Galileo and his successors lay in showing that forces are accelerations by which a state of velocity is gradually brought about. The words cause and effect still linger, but the old conceptions have been dropped from mechanical philosophy; for the fact now known is that in certain relative positions bodies undergo certain accelerations. Now an acceleration, instead of being like a velocity a relation between two successive positions, is a relation between three; so that the new doctrine has consisted in the suitable introduction of the conception of Threeness. On this idea, the whole of modern physics is built. “

Posted by: Jonathan Vos Post on March 2, 2008 8:09 PM | Permalink | Reply to this

### Re: Peirce on Mathematics

As usual, “threeness” is unstable here. It’s stable to talk about cause and effect – duality – so much so that we still use that language. But once we move to three, why not four, or five, or… why stop just with accelerations and not relate more and more points?

Categories, with their two levels of objects and morphisms, are stable concepts. But once we open the door to bicategories with their three levels, we can’t help but consider four, or five, or… why stop there?

It seems that once a pattern is established at “three”, the inclination is to continue it on to infinity. Ones and twos we can accept on their own, but not threes.

Posted by: John Armstrong on March 2, 2008 9:13 PM | Permalink | Reply to this

### Re: Peirce on Mathematics

I can maybe see a family resemblance between situations of firstness, and equally for secondness and thirdness. But Peirce suggests that the connection is much stronger.

For Peirce, thirdness appears in mechanics because accelerations relate to three of a body’s positions, as shown by differential equations being of second degree. So, were there no third, or higher, degree differential equations being used in the 1890s?

Or is it possible that one can play the trick of analysing something higher order as a composite of lower order things? Peirce himself does this for four place predicates, showing how thay can be analysed into ternary ones. E.g.,

— sells — to — for —

can be rephrased as a complicated combination of ternary relations.

I know a little about attempts to express functions of several variable in terms of those of 2 or 3 variables.

I wonder what Peirce would have made of n-categories. Would he have expected three layers of structure to have sufficed?

And a question for us, what do we expect is shared between all situations which can be represented by a n-category, for a given value of n?

Posted by: David Corfield on March 3, 2008 10:29 AM | Permalink | Reply to this

### Re: Peirce on Mathematics

I have one amateurish thought on threeness that occurred to me long ago when I first read about Peirce. As far as the buildup of usual spaces are concerned, there is something quite fundamental about three. The most elementary manifestation is when you have a collection

$\{X_i\}_i$

of spaces, subspaces

$\{Z_{ij}\subset X_i\}_j$

and isomorphisms

$\phi_{ij}:Z_{ij}\simeq Z_{ji}$

When can one use these to glue the $X_i$ into one space? Exactly when the condition

$\phi_{jk}\circ \phi_{ij}=\phi_{ik}$

is satisfied. This has always seemed rather nice to me, that it suffices to check consistency of triple overlaps’ and need to go no further. Similar things occur of course with glueing conditions for sheaves on a spaces. Perhaps the fancy way to say this is that 1-stacks in groupoids are very much a part of nature, and there, threeness is fundamental.

This is a very off-beat remark as far as Peirce is concerned, but since n-categories already came up in this connection, I allowed myself to insert it.

Posted by: Minhyong Kim on March 5, 2008 11:03 AM | Permalink | Reply to this

### Re: Peirce on Mathematics

Similar things occur of course with glueing conditions for sheaves on a spaces. Perhaps the fancy way to say this is that 1-stacks in groupoids are very much a part of nature, and there, threeness is fundamental.

But for sheaves (= 0-stacks) it’s twoness, isn’t it? In general, for $n$-stacks it seems to be ($n$+2)-ness which is fundamental, in this sense.

Which, by the way, may be regarded as following as one tiny aspect from the immensely general theory of “nonabelian descent” of Ross Street, which we were talking about in the other thread.

Generally, I feel that 1-stacks are receiving an undue amount of attention and appreciation relative to $n$-stacks for other $n$. But that’s just me.

Posted by: Urs Schreiber on March 5, 2008 11:33 AM | Permalink | Reply to this

### Re: Peirce on Mathematics

I sympathize with your view. This is why I referred to usual spaces.’ In this context, I would say that if a space is given, then maps to it involve two-ness. But the data for the *space itself* involves three-ness, and stops there. Of course you understand this much better than I do.

Perhaps it’s fair to speculate that in Peirce’s time, some concepts involved in the construction of usual objects may have informed his ideas at some level. But perhaps not higher-order objects.

Posted by: Minhyong Kim on March 5, 2008 12:32 PM | Permalink | Reply to this

### Re: Peirce on Mathematics

Urs wrote:

Generally, I feel that 1-stacks are receiving an undue amount of attention and appreciation relative to $n$-stacks for other $n$.

The lowest-dimensional high-dimensional things always get undue attention. They’re the easiest to understand, and the first place where some interesting phenomena start showing up.

There was once a conference with the amusing title “Low-Dimensional Topology and Higher Categories”. What’s low for topologists was high for category theorists.

Posted by: John Baez on October 13, 2009 9:04 PM | Permalink | Reply to this

### Re: Peirce on Mathematics

When people ask me what kind of maths I’ve been doing recently when not thinking about penguins I sometimes say “low-dimensional category theory”. I grew up doing low-dimensional topology, which means up to dimension four or five; so I think of monoidal bicategories (braided, symmetric or otherwise) as being classically in the realm of low-dimensions.

Posted by: Simon Willerton on October 13, 2009 10:08 PM | Permalink | Reply to this

### Re: Peirce on Mathematics

John wrote:

Urs wrote

Generally, I feel that 1-stacks are receiving an undue amount of attention and appreciation relative to n-stacks for other $n$.

Oh, wow. When did I write that? Ah, that was one and a half year ago!

John writes:

The lowest-dimensional high-dimensional things always get undue attention.

I can’t argue with that. I forget what I was thinking back then. Looking back at the discussion it seems the point was less about attention actually than about relevance.

The argument I was reacting to was of the kind: this and that is special because that’s the way it works for 1-stacks.

Posted by: Urs Schreiber on October 14, 2009 10:16 AM | Permalink | Reply to this

### Re: Peirce on Mathematics

DC: I wonder what Peirce would have made of n-categories. Would he have expected three layers of structure to have sufficed?

There are indeed several places where Peirce asks the question whether a specific process of reflective abstraction under consideration might exhibit a limit — or at least a periodicity that limits its capacity to continue generating radical novelty indefinitely.

For my part, I have run across a number of puzzles in combinatorics that I connect with this question, though I’ve never gotten very far in resolving them.

I’ll collect some notes …

Posted by: Jon Awbrey on October 14, 2009 12:28 PM | Permalink | Reply to this

### Triadicity : Reducible and Irreducible

Here’s a bit on The Importance Of Being Triadic.

NB. All my heroes have always been cowboys and all my proofs have always needed checking.

Posted by: Jon Awbrey on October 13, 2009 5:14 PM | Permalink | Reply to this

### Lasso Knot Theory on the Domain and Range; Re: Triadicity : Reducible and Irreducible

All my heroes have always been cowboys
And still are, it seems,
Sadly in search of, and one step in back of,
Themselves and their slow moving dreams.

Or should that be:

All my heroes have always been theorems
And still are, despite any spoofs
Sadly in search of, and n+1 steps in back of,
Themselves and their slow moving proofs.

Posted by: Jonathan Vos Post on October 13, 2009 7:33 PM | Permalink | Reply to this

### Lasso Knot Theory on the Domain and Range

Best Laff I’ve had all week —

Ja Ja

Then again, it’s only Tuesday …

Posted by: Jon Awbrey on October 13, 2009 8:24 PM | Permalink | Reply to this

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