Guy_ in_ snow.jpg

Frigid dustlets of snow whipped around his ankles while he waited outside for the smoke to clear his lungs. He hadn't bothered to put socks on and errant particles of the snow managed to seep into the tops of his boots. His face had traces of expressions, but none lasted long enough to give him the appearance of any particular mood. The sound of the television was loud in the background and it helped him remain outside a while longer, even after his cigarette had hissed and died.

***

Reply

I started all that
you just copied my thought
and changed it to snacks

-18th century, Anonymous' husband

Snack based haiku

waiting for burkey
feels like an eternity
i eat a sandwich

-18th century, Author unknown

Hasten to join a rascality of jugglers, my good man.

“But in a better Age, when Learning flourished more among the Romans, the Mathematicians were in greater Esteem, and Cicero carefully distinguished them from that Rascality of Jugglers that arrogated to themselves the Foreknowledge of future Events” [Isaac Barrow, The Usefulness of Mathematical Learning explained and demonstrated, Being Mathematical Lectures read in the Publick Schools at the University of Cambridge, 1734].

"The literature of the discontents"
"The literature of the malcontents"
"The literature of the content"
"Memos to the boss"
"Emails about the boss"



Correctness, rectitude, rectify, regret, regress

Understanding the nature of the mechanisms lined up in front of you is a worthwhile way to occupy your time in the event that one or more of those mechanisms should have to be altered, replaced, or destroyed. There are limitations on the tolerance of people for certain forms of domination, coersion, or the shaping of their lives by others (these limitations fluctuate and are relative to the historical scenario at hand: they are lower and higher in general in different periods, and also relative to the perhaps historically inexplicable variations in the individual natures of the people who make up the body to be analyzed and the relations between those people). There are also limitations on the ways in which people can react to the movement of the previously mentioned limitations in either direction (either towards a more palatable and preferable state of affairs or towards one which they are less inclined toward). Understanding the limits which affect the ways in which one can alter or react towards one's political situation is one method which can (sometimes) enable the alteration or expansion of those very limitations. The intolerable situation imposed by, e.g., the inability of women to vote was obviously the motivating factor for the emergence of the women's suffrage movement, which effectively altered the ways in which a woman can react towards her political situation. Certainly a great deal of apathy preceded the emergence of this movement, and it is unlikely to have been predicted 200 years prior to its emergence, but emerge it did. Similarly, today there is a great deal of apathy in the West towards a certain cluster of problems which are 'taken seriously' by academics insofar as they write about them but about which very little is done on the ground because the feeling seems to be that certain things might as well be accepted, given the seemingly insurmountable difficulties which have to be faced in order to alter the situation. Or, worse, the solutions to the general problem hold no intrinsic interest because the problems, such as they are, are only understood as they apply to the individual at hand--the academics write as if the situation were otherwise, but this writing is also bound up with the whole apathetic turn connected to the tendency to understand problems in terms of the individual.

But, just as women's suffrage was by no means an 'obvious' solution to the felt-dissatisfaction of women and men prior to the emergence of the right to vote for women, similarly, we can't imagine what possible solutions might emerge from the perceived miasma of apathy infecting a large portion of the population with regards to the ills of, e.g., bureaucratic civilization or capitalism. All the big successes of the world begin small and unpredictably, and it is only with the hindsight of history that we are able to describe the logic of their emergence as if it were something fated. That the academic's gesture towards communal action and things of that sort are all but hand-waving and journalism is made apparent by the way in which those selfsame academics live their lives, for the most part, according to 'individualist' criteria. This may have the result that a meaningful solution to "the problems" in the large sense will actually be a rephrasing of those problems in a much smaller, individually more soluble sense. Or, it may have the effect of becoming such an intolerable contradiction that new forms community based activity arise which are not based solely on limp academic proscriptions. Another option, too, is that there may very well emerge something new under the sun, wherein the seeming opposition between the individual and the community is supplanted with something entirely different, which cannot at all be predicted or completely understood from within the stock of concepts currently available to us. One thing is certain, though, and that is that things will not simply stay the same; the world will not simply reproduce itself infinitely, small accretions have an additive effect and, over even the relatively short time span of 50 years, much of the political landscape of the world will likely be completely unrecognizable by all but the most schematic of understandings. But, too, our lives are generally short in comparison to the time span at work in deep and meaningful political change (though this seems less true relative to the rapidity of change in the last 200 years or so). So it is not at all a surprise to see many people eager for change throwing up their hands in disgust and resignation at what will most likely be another decade or more of roughly the same kind of apathy, and in all probability longer than that unless something remarkable happens (as is occasionally the case).

The least requirement, in any case, is in being genuine and fighting the ironic or distanced engagement prevalent as much as one's temperament allows; and often, as in my case, this is not very much at all.

Pulse pounding brain beat


Manganese

Who needs anything when you can have everything.

the snack window!

Little Bits

The remarkable thing about astronauts

impermeable, silicate

salt-stain

A thing meant

the brutal preamble

From concrete, to abstract, to concrete, to abstract, and all the points between every time we talk

"The word "viscosity" derives from the Latin word "viscum" for mistletoe. A viscous glue was made from mistletoe berries and used for lime-twigs to catch birds."

Motto

"An ontological analysis of things simply called 'mathematical objects' is likely to miss the real point of mathematical existence" -Saunders Mac Lane ("Structure in Mathematics" [1996], p. 182).

"Yes, there they are, still making comments about themselves"

I feel like God would say that if he woke up and took a look around Earth for a few minutes.

earrings

Long walks on the beach.

Constancy

I'll change my clothes.

And still be the same.

coats

nothing much

to think

just blood drained

thinking lately

the rest of my life

no rhythms

point sets

no neighbourhoods

100 to 5 in three or four days

a week at the high end.

all that kind of thing

words wearied into the wind

that kind of thing again

hello fair world

still the same fucking fool

isolated dendrites

clucking their idiots

into my idiot face

he had a face like an idiot

the idiot face of an idiot

the stupider the smarter

everything is forever

Takes

refrains and the chorus and rondels, arpeggios repeated into the body


one of those fuck you types
swear word afternoons with mint juleps and tongue compressed linen suits linked to the flesh with the sweat of the flesh.

fuck written on the floor boards
but you look at the ceiling for the day

"It's so sensitive, it needs air, it needs to do some living"


busy being grown up, forgot how to smile.
can't talk, ear against the wall.
can't hear, deaf.


"No one's ever escaped"

just
look

ok

Nothing can bridge the gap.


"I disagree, there isn't one."

"My grand-daughter and her boyfriend who won't marry her are going to become art."

"It was years before he realized it"

"It was years before he directed it"

'realisateur'

the sooner you see that




Yeah, keep talking.
(That works).

Chill and shake

Face like a thresher, it just tore everything to pieces.

(You need a separate baling face for stacking the pieces into manageable bales)

Song 1

bitter bitter young man
bitter bitter young man
bitterrr

younggg

ma aa aaaann

(in sweet sweet voice of woman, accompanied by snare drum played with brushes and banjo, slight, in the background)

a little bit more every day a little bit less every day
every day now

every day every dayyyyy now

These things are only difficult because it's easy.

Telegraphy

Face like a salt stain.



Hair like a mastiff.



Eyes like an oil tanker.



Hands like a suicide.



Face like a salt stain.



Run it in the rain.



Leg it, hoof it.



Get the whole thing done.

Little Guys

I keep remembering and forgetting and inventing things that I was and am, it's irritating today but usually it is alright. Living a life or whatever is the map of a point made by running your fingers beneath a loosely stretched infinite sheet of silk. Happiness is those rare occasions when your finger becomes bloody and raw and the life of you seeps into the fabric and you remain everywhere a while, until the blood decays and blackens and crumbles.














There are five greyed pieces of driftwood and all of them angle in unknown currents, salted and cracked from years at sea. There are five pieces of driftwood and each of them is discovered one by one, snared and meandering (noose-like) in the currents, of the purposeless sea. Five pieces of driftwood all end up choking idle in the shallows, five all end up wrecked on weird and ugly shores, remembering what they were.















Advice: Don't pollinate time with your angry insides.
Advice: Your stomach is smarter than your brain.
Advice: The slowdowns come too late to matter.

Life and death in the dust

Floating around and
bustling, always,
always in my own skin cells.

Sitting stiff and stoic,
waiting for meteors and diseases.

(They don't come).

Expected strands of
mid-morning half light,
with their heated, glancing blows
shuffling me awake again,
stuffed and mounted again.

And the motes whistle around,
and they settle down.
And we settle in,

and clench our teeth for the long haul.
Grinding the days and the enamel down,
both, to the bone.

Such narrow means to shape.

Nights spent sweating tangents alone.

No stranger to them, but they're still strangers to me.

Yeah, there's something wrong with me.

starting to feel it


































































yep.

Gods the love and the light

even the fibres of it

thin and true

late night


like this

a little bit like this













i dont even care what you do

or what i imagine





i dont care



























i dont fucking



care

i abandon everything to everything
because what there is
is what you are

you
and the savage savage thing of it

the brutal hurt
and harm

"What am I to you?"

"You? You're a disappearance."

She growled silently in the fog, and he clenched her hand with the cold weight of a long walk and a long life, both stuck together, clammy and final.




"Oh stop with the literature," she said finally.

He laughed. "I'll stop with it once we stop being characters in my mediocre novel."

"Yours?" She spit it out. They didn't speak for hours.

Hoo boy what a night.

The kind of things that history brings to your lap like a dog with a saliva soaked paper that you never wanted to read.

Imagine graphing the absorption ratio of particular letters set in times new roman when they interact with particular kinds of dog diets.

"The dog ate a snail today, those Ns are in for a world of hurt" That sort of thing.



You know, I've been straying from the path a bit lately. I remembered today.

I really do hate every person on the planet.

Except for you, of course.

You, you I love. I love you like the way I can't stand being stuck in the bottom of a well, or pinched at the waist in a cave with spiders in my mouth.

I love you the way I can't even tell it to you because the barrage of bare Burke is enough to cave your fucking life in ten hundred times a second and what am I supposed to do then fine fuck. Alright.

Alright.
Yes. Calm down. Say yes to things and not no to things.


Don't even belief in a thing.

Just philosophy a thing and ask "what is a thing" that sort of thing, you fuck.

Yes yes yes. Ask about life instead of be in life, that sort of life. Alright.

What kind of relationship do you have to your own thought, what sorts of things are worth caring about, or, what sorts of things should you ignore.

Alright.

Check please.

I had a dream last night where I killed myself out of spite; my sister had been doing something (I forget what) that I couldn't stand, and I had that impossible-dream feeling where you can't make something happen until you do another thing, and the two aren't all that connected. Anyway, to alleviate the feeling I jumped in front of a train and killed myself. It was the first time I remembered dying in a dream. It was awful and I felt awful when I woke up, but I knew also that I deserved the feeling. Then, 5 minutes later, I was all like "yes, you are an awful person, but... you only think you deserve it because you still believe in sin" and then, five minutes after that, I chastised myself both for my iconoclasm and for my lack of contrition. Anyway, I'm scared to sleep now because I think I've killed my dreams and sullied the good things in me which were left unsullied (there weren't many probably).

I wish I could say I love you in that brutal savage way that I know is waiting and lurking in the corner parts of my veins. But, instead, I say it and it is the song of my weakness, weird and soft, callow, fickle, all of those things alright. One of these future days I'll force it out of you with brutality and the husk of me will molt. Probably probably.



These long days I keep reminding myself that the seconds I steal from you are still the precious ones and that I'm living a second life like a phoenix because everything that is, now, shouldn't really have been because of what I was. A few years lost to idiocy and fake fake fake things, and now scrabbling back into it. Alright fuck.


I feel like a weak man both in the ways that I give in and in the ways that I resist.

What kind of ways are left for me.

What does Buddha tell you to do when you've taken the middle way and it leads you to the same? Oh, right, Buddha doesn't 'tell' you anything. He tells you what you know: life is suffering. And, then, he hints at what can't possibly be true: that it's an illusion. Life wouldn't be suffering if it was an illusion, it's the reality of it that makes it what it is. It'll take another two thousand years for religion to present suffering without exulting or denying it.


"There are things."

What kind of religion is that?

The well ain't deep enough

Covered up teeth marks.

Bergmann

"There are functions. Of course there are."

Spray paint just the spray paint yes one two.

THE RIGHT KINDS OF BEAUTIFUL

and

THE WRONG FUCKING KINDS OF BEAUTIFUL

sometimes music just fucking makes life make sense

god

yes

Night times

"Got that sweet pain ache"

feelin' so good, even with them eyeballs bulgin'


we never learnt nothin' that a body could forget

So much in my brain I cannot breathe

I had a thought today on my way to school from our brief conversation last night about language.


Before, I had been imagining that whatever structurally sensitive way of doing things I came up with would be a way of overturning or at least expanding the broadly object-oriented way of thinking that dominates most conceptual thought. I usually call that way the 'thing language' because that's a phrase that Rudolf Carnap uses and I like it. Anyway, today I've been thinking that my project should more be to show that the thing language (any thing language, I would guess) is already intermingled in unclear ways with structures and that the real problem is that the vision of a language restricted to things just doesn't make sense, and that we are just misled by the idea that some languages can (i.e., set theory) can just have things. I guess that is half what I've been doing but I was thinking mostly that the structure thing I would offer would be somehow embedded still in the more basic thing language and that is what I've wanted to avoid. I guess the way to do it is to show that that language itself isn't what we usually think it is. And, come to think of it, all the troubles that set theory has with paradoxes and everything like that, and all the problems that Quine has when he tries to make meanings into just arrangements of things, all of that stuff points the way to a hybrid view which has 'things' and 'structures' both, but in a way that somehow advances beyond Frege's troubled and somewhat confused division between objects and concepts. But that 'somehow' is the hard part, and figuring out what it is that is troublesome about Frege's view is also difficult.

So anyway, I guess I'm a bit worried that I don't really have much to add from Frege's stuff other than specifications. I think what I will do is a kind of focused examination of Frege's views on logical simplicity and the absolute quality that he gives to the distinction between concepts and objects. I would like to do something like what Bergmann does when he reads Meinong. He shows that Meinong's theory tries to do something (i.e., explain relations in terms of objects) and he also shows that Meinong's way of doing it pushes that method to its absolute limit but still fails. I think there is something like that going on with Frege, that the use of the thing language just forces him to have this completely unclear mess of conceptual stuff that is difficult to make any real sense of given the rest of his doctrines. So, instead of turning to Bergmann and Meinong in the later part of the thesis, after I've discussed category theory and structuralism and all of the mathematical history stuff, I think I will return to Frege to show that all the essential problems (the 'deep' ones) are already there, and usually in a clearer form than their later more jargony incarnations.


So thesis will look something like this:



-History of geometry up until Hilbert, to illustrate that Hilbert begins structuralism (or popularizes, whatever)

-clash between object-oriented Frege and radically relational Hilbert

-increasing intermingling and confusing of object/structure distinction in set theory, leading to development of category theory and structuralism as a philosophical position

-discussion of structuralism, particularly with reference to ontological matters (i.e., debate between nominalism and realism)

-discussion of category theory as a thoroughly relational view which does not necessarily possess the early drawbacks of Hilbert-type structuralism (argue against Solomon Feferman's claim that category theory is just like set theory on this score by suggesting that I can provide a philosophical basis for the structure concept)

-return to Frege's division between concept and object as a fundamental matter, show all the problems with it (e.g., concept horse) and argue that the problem for structuralism as I see it is that people take the Feferman line and worry that we can't make any better sense of 'categories' than we can of 'sets' once we get outside category theory and set theory; i.e., that we can't provide any serious philosophical backing for the notion of set or category or whatever.

-do something like Mendelssohn does; explain what the problems of Frege's view are and try to clarify them, but show that the real problem is the embedding of concepts construed as objects into an already object-centered way of picturing thought, show that we have to alter our picture to show that what we actually do involves mappings and morphisms and all kinds of other structural notions, even when we deal with objects. Don't try to replace all objects with structures though, I think there is something fundamental about the distinction and just using one instead of the other seems like no progress at all (even if there is a lot to be said about the view that objects are structures in, e.g., physics).

a world submerged

The wheels were slowed by clumps of moist soil and dead blades of grass

His sandwich eventually arrived, and all he'd managed to write was 'Dear Angela,'. He pulled the sandwich apart and ate it's constituent elements methodically, imagining the order to be very important though he didn't bother to explain to himself why. He ate it and idly pictured himself an anachronistic fop dappling the corners of his clean-shaven face with the bleached white napkin. The napkins here were the cheapest, thin variety and the acid of his saliva combined with mustard easily cut through and created ugly, ravaged-looking holes. He saw bits of the napkin getting caught in his beard, rolled up with mustard stains and glued on by his spit. The thought disgusted him and he went to the bathroom to wash his face. He returned and sat, stirring his second coffee and staring ahead ahead. When he finally finished his lunch, he weighed the pen in his hand and closed his notebook.

***

Two years later he met Elise. He never bothered to write to Angela, never returned to his apartment or to the memories of the grotesque sexual carnival that their grey and pale bodies had performed for the appliances there. He never returned to work, never collected his paycheques, never wanted for money. He could barely recall how he had met Angela now. When it came down to reconstruction, he had settled on a brief workplace courtship as their official history.

His fondest memory of that period was the easy way in which he had dissolved out of their lives. Since then, he had hadn't so much as imagined their reactions, her face, the way her eyes leaked, red for too long after the mood of him had passed, the sick desire she'd had for the corpse of him. Only now did he cast his mind into the faint glimmer of family and food they'd made together, however briefly. And even now when he looked back it was always with indifference and benign amusement.

Mostly, though, he thought of Angela's father gazing aimless into the space between his wife and daughter, a source of echoes like the walls of an abandoned quarry.

Mostly he just didn't think at all.

A series of accidental women had led him to his spare and barren apartment, to the vapid world of art. The strange ease with which he produced wealth encouraged the increasing disconnect between himself and his history that had begun well before he'd met Angela. He'd tangled himself into the city and mapped all its alleyways and atrocities. Imagined the city and himself, and more than imagined, more the impostor every day. His personality slowly took the form of a burial into tiny ceremonial relationships, little set-pieces, locked away and catalogued to be forgotten. For a brief instant each one of these was perfect and beautiful in exactly the way his thoughts about Angela had been. Sometimes this feeling lasted long enough that he could discern an imagined interior for the faceless bodies of the bodies he employed for love and affection and friendship. But these moments were few, and grew increasingly rare. He had become a collector of roles, keeping everything of himself in tiny self-contained boxes. No overlap, no history, just individual beautiful notes with no rhythm and no melody. "The cold music of fact," he thought that sometimes and smiled. He'd become something of a quarry himself and he smiled at that too.

Two years later, though, he met Elise and his careful segmentation started to blur into a new kind of history.

***

His name was Peter. Every day he woke up and beside him was his wife, whom he loved, and, on most days, one of their two daughters had bullied their way into his consciousness with a request (a demand). No.

His name never mattered, and today he found himself alone in a room whose dimensions he couldn't quite make out in the half light. It was clean and cold in there, and there was an air of violence. He tried to sit up but was somehow constrained, though he couldn't feel any straps or restraints of the usual kind. "Welcome," a voice intoned.

No, today his name was Jonathan again and his head was pulsating with horrible pain. Someone, not Elise, stroked his forehead gently and whispered strange words in a language he couldn't understand in his ear. The skin along his shins was raw and bruised, and he imagined walking would be a painful ordeal. "Thank you," he said to the unfamiliar woman. "Thank you". And he let himself weep.

***

The phone rang and he knew it would be Elise.

He dragged himself away from his desk and picked it up on the third ring.

"Hello"

"Hi." [It wasn't her. He panicked.]

It was her. They were supposed to meet for lunch but she wanted to cancel.

"Lots to do at work?"

"Not really," her voice was flat, "I just don't feel like going anymore, I'm going to finish up here early and come home."

"Bad day?" he asked.

"No, not bad, just," her voice didn't trail off but there was no commitment to the thought shaped by her sentence and she abandoned it. This was odd for her, he thought.

He conjured up worry, felt it. "Look, I'll come pick you up, come home and take a nap. Don't think anymore"

"I'm not thinking now. Don't come pick me up, I'll just finish and come home on my own, it's alright. I just didn't want you wasting your time coming to lunch when I'm like whatever it is I'm like."

"Well, okay, just call me if you want I'm sitting here pretty whatever you are myself, I've been drawing a circle for half an hour."

She laughed, and they agreed she'd come home in a few hours.

How many men it took

Life is long enough that nothing you do matters at all except everything you do.


It's short enough, too, that for most of us the circle of history that spreads out from the little ripples we make eventually ends and we are swallowed up into dust that turns into smaller dusts and eventually into balls of energy that collide and crack apart and cease to vibrate entirely. And then there is calm and the looping of calm indefinitely until the whole thing crushes back into itself and we start the stupid dance again, making the same good choices that lead us to love in the first place.

Dark delicates.

That is both a nice and an unpleasant thought, mostly a nice one. Mostly a nice one these rain day.

Oh life you wily one.

Life you wiliest one of ones.


(Think really tiny thoughts at night and really immense ones in the morning and nothing in between, except snacking).

Retrograde

the ornate lies of ptolemaic epicycles

piling up until the final loose strand breaks things irreparable

just destroy from the start

and it's never a problem

My gal is red hot, your gal ain't doodly squat.

Kill and cook

aimless aimless hate

but well placed loves


hallmarks of good life
a good life


degrading orbit

crash to the depths


and you got no lid

The tourniquette slowly loosened the further they walked up the mountain, and little bits of dust and filth started to cling to the drying blood.

All that morning he came up with things to say to her that might force her into his life, but the reality he wanted remained just on the edge of his vision, like a thin mist that disappeared once he was in it. He wandered around the streets, bought a coffee and drank it absent-mindedly, burning his tongue.

He ended up lurched over a notebook, another coffee in front of him, at a diner he normally would have avoided. The food was too expensive, but looked like it was too cheap. His pen was placed neatly parallel to the right edge of his notebook and he watched the cream swirl into his coffee, Brownian motion and the spin of the earth combining to mix it so that he didn't have to move or think.

In the drift of feet that had brought him here, to this glossy table and plastic smelling air, he'd decided to write her a letter, detailing the truth of things as they were, in a clear staccato style. But now that he was here, all he could imagine were the trillions of coffee molecules bumping into each other and intermingling with the milk and the air, exchanging heat like hellos. The initial drive he had to bare everything for her had dissipated, so he tried to conjure the violent things he imagined when he saw the shapes that comprised her face.

[He imagined burning her house down, for the hundredth, thousandth time. Watching the cracks of despair visible from the top of her head as he pressed her face against his chest. The firefighters arrived and the blaze was tamed, but all that was left of her apartment was a gutted remnant, pools of water forming in her closet around her mementos and knick-knacks.

That night they fucked in a cold hotel with the sounds of the highway loud in his ears. They'd turned off the lights but the green glow of the parking lot seeped in and made her eyes look like and skin look like wax. He imagined a thin-legged insect landing on her, and her incapable of precise enough feeling to notice. They fucked for what seemed like hours, and for the duration he tried to pull himself out of the machine-valve feeling that he always had with her when they did this. But nothing worked, and he made the right moves by rote. Eventually she made signs of satisfaction and he rolled off of her, bits of sweat forcing the starchy sheets to cling to the back of his calves. She gripped him panting and tried fumblingly to continue, to pleasure him, but he rolled away from her and mumbled some incoherent nothing which she didn't pursue. A smile leaked across her face and momentarily he recognized the makings of an expression on her face. But it passed. The sheets clung to both of them and between them and he was mildly happy for the barrier. He was happier still that he'd destroyed her home and her past and here she was, soft and molten beside him as if nothing had happened.]

The thought of her drenched in anxious sweat, satisfied and dulled by him, brought him back to the letter and he picked up the pen carefully while he waited for his over-priced salad.

Despicable Face

Furniture face, knife face, clockwork face, disease face, stealing face, rapid face, hidden face, despicable face, romance face, line face.


Yeah but only the good faces from now on.

Well, and the mean and funny faces, too. (small times for them too)

If I had a voice, I could sing.

"microscopic circus in the fluid of my eyes"

There was a slight fracture just below the waterline, and the salt would seep in and crystallize in beautiful patterns for her

A week later and everything remained still, the trajectory of new things indistinguishable from that of old things. Occasionally he had imagined that throwing himself headlong into a life with her would have changed something in him, perhaps made the machinery of him into a connected whole, capable of seeing faces and understanding gestures in the proper way. But none of that happened. He could barely bring himself to pronounce her name. 'Angela'. He imagined the roiling grease of her name and its constituent syllables leaking out of his mouth and emitting a foul smell as it slid, writhing, to a bare pine floor. When she was asleep and nameless he wrote out their futures together and everything seemed rich and promising, but her eyes and the confusion of her features inevitably interrupted and only he remained.

On Tuesday he woke up early and decided not to go to work. He sat on the edge of the bed with his right hand stretched behind him, testing the tensile strength of the skin along her torso. He pressed too rapidly on one of his forays and she woke up, smiling.

"What are you doing awake? It's six thirty, come back to sleep."

"I'm going out for a walk," he said. He leaned back and kissed her hair into her forehead, a few strands pressed down by the heat and moisture of his breath. And she smiled, too, still half asleep. "I'm skipping work today, it's nice out."

"I can't call in sick today, you shouldn't either."

He ignored her and got up to get dressed. His mind was fixed, he was smiling now, a strange smile.

"I can't go to work today, I need to get out for a bit. I'll call you for lunch, we can have lunch together."

She looked at him, somewhat puzzled, but the moment passed. "Alright, if you have to! Let's meet at 11, I'll take off for lunch early, ok? Call me! I'm going back to sleep!" She spoke rapidly and there was something sad in the light way she'd accepted his leaving, but this feeling, too, passed.

***

His shoes were tight around his ankles, he'd chosen too-large winter socks for the surprising warmth of the early spring day. The sun glinted into his eyes and he walked for blocks with them half closed, imagining where to go and finding no resistance in the early morning streets.

Chapter 1 and Chapter 2

Chapter One: Frege and Hilbert on the Nature of Axioms

Basic aim and outline of the chapter:

The aim of this chapter is to establish some elements of the ontological framework within which I wish to situate my later discussion of the concept of structure. In my research I have found the correspondence between Gottlob Frege and David Hilbert to be a crucial step in the historical development of structuralism within mathematics and the philosophy of mathematics. Accordingly, the chapter begins by recounting some of the conceptual elements of the history of geometry which led to the 1899 publication of David Hilbert’s Die Grundlagen der Geometrie [The Foundations of Geometry]. With this short historical portion I hope to illustrate the importance of Hilbert’s view for modern mathematics, as well as its radical divergence from a traditional view of the nature of geometry. From these historical remarks I then move on to a more detailed discussion of the content of Hilbert’s views regarding the nature and meaning of his axioms and definitions. I then move on to an evaluation of Gottlob Frege’s criticisms of Hilbert’s views, criticisms which have largely been ignored but which are, I believe, quite helpful for understanding both the historical development of ‘structuralism’ within mathematics as well as several of the conceptual problems involved in structural approaches in the philosophy of mathematics. The chapter concludes with the tentative suggestion that Hilbert’s later, formalist work began to take Frege-type criticisms more seriously as the philosophical difficulties of his initially radical position became clearer; I attempt to substantiate this suggestion in the second chapter.

Summary of the chapter:

After some very brief remarks regarding the historical status of Euclid’s Elements (with emphasis on problems with the parallel postulate), I move on to outline some of the conceptual shifts which occurred within geometry after the development of non-Euclidean, hyperbolic geometries by Lobachevsky and Bolyai between 1829 and 1832. My main claim here is that these new geometries, which were significantly extended and generalized a few decades later by Riemann and Gauss (among others), eventually created a widespread philosophical confusion about the nature of geometry itself. Prior to the development of non-Euclidean geometries, the generally accepted view was that geometry was a concrete science which helped to explicate the laws governing the actual, three-dimensional space in which we live. A commonplace epistemological adjunct of this view, emphasized and popularized by Kant, was the belief that the axioms and postulates of Euclidean geometry were truths that we can grasp immediately through our spatial intuition. These intuitive truths neither require nor admit of any form of proof, they are basic.

Beginning in the 1870s, when the work of Riemann, Helmholtz, Gauss, and others began to seem scientifically important in ways that the Lobachevsky-Bolyai geometries had not (or at least not immediately), the traditional philosophical account of the nature of geometry no longer seemed adequate. It is by no means intuitively obvious, for instance, that there are or can be points at infinity which guarantee the ‘truth’ of the axioms of projective geometry, nor is it obvious how to develop higher-dimensional differential geometries based solely on the deliverances of our spatial intuition. Perhaps least intuitively graspable among the new geometrical ideas was Riemann’s conception of a space whose curvature is different at each of its points. The full generality of the new forms of geometry was revealed by Riemann in a lecture (delivered in front of Gauss as his Habilitationsrede in 1854) entitled ‘On the Hypotheses which underlie Geometry’. In that lecture Riemann discusses five hypotheses which lie at the center of geometry, which he proceeds to treat with a level of abstraction and generality almost totally absent from the traditional Euclidean understanding of geometry. In this lecture, published only in 1868, Riemann provides clear and general formulations of metric and topological properties, with corresponding metric and topological ‘spaces’ determined by these properties. Despite the sophistication and facility with which Riemann and others employed their new geometrical concepts, there was no philosophical correlate to their work. Whereas the traditional, Euclidean view had a considerable philosophical edifice supporting it and linking it with the wider world, there was as yet nothing like this for the new approach to geometry. The conceptual shift remained an almost entirely internal development of mathematics itself. Of course, the lack of a firm philosophical understanding of the nature of their science did not dissuade geometers and other mathematicians from pursuing these deviant geometries, far from it. The second half of the nineteenth century saw an unparalleled explosion in geometry despite the lack of conceptual clarity regarding the nature of the various geometries themselves. Differing opinions as to what was going on within geometry eventually arose, however none of them met with much success until a general consensus became somewhat guaranteed by Hilbert’s 1899 work.

Following this portrait of the historical lead-up to Hilbert’s work, I will turn in earnest to the content of his views. My initial aim in analyzing Hilbert’s views on geometry and geometrical axioms is to illustrate how radically different his approach was from the traditional Euclidean/Kantian view of geometry. Though Hilbert paid lip service to the belief that his axioms, too, were grounded in intuition, in practice intuition plays a very minor role in his system. Instead what we find with Hilbert is a disconnection between the axioms and their fixed reference to an actual, concrete system like three-dimensional Euclidean space. While much of the terminology in Hilbert is familiar from Euclid’s Elements (he retains the terms ‘point’, ‘line, ‘plane’, etc), the meaning of this terminology is radically different. Hilbert’s intention, as he makes clear, is to describe an abstract structure which can then be interpreted in any number of appropriate concrete instances or systems. His view can be seen as an early version of modern day model theory, in which the uninterpreted elements of the abstract structure are assigned concrete elements by way of an interpretation. If all the axioms remain true on a particular interpretation, we call this interpretation a model for that set of axioms. (Hilbert’s terminology is of course slightly different, but the basic idea remains the same). In this way, Hilbert employs an abstract structure to prove claims about the concrete Euclidean system; in other words, he constructs a set of axioms for which the concrete Euclidean system, appropriately interpreted, is a model. He proves, for instance, the consistency of his axioms by showing that there is at least one model satisfying them. He also proves the independence of certain axioms from the others by providing interpretations on which the ‘independent’ axiom is false while the others are true. Especially interesting for my purposes is the fact that Hilbert takes the consistency and independence proofs for his set of axioms to imply the consistency of Euclidean geometry itself, and the independence of certain axioms of Euclidean geometry from certain others. This type of approach, as we shall see in the second chapter, was wildly successful, so much so that it is difficult to imagine the history of mathematics, logic, or even formal semantics in the twentieth century without the model-theoretic apparatus which is germinal in Hilbert’s work. It should also be clear, however, just how radically different this view is from the traditional Euclidean view of geometric axioms as intuitive truths about three-dimensional space.

From Hilbert’s views I move on to those of Gottlob Frege. Frege’s own views on the nature of geometry were largely ‘traditional’ in the sense outlined above. He was in several particulars a follower of Kant, and he considered geometry to be a concrete science whose basic truths were immediately apprehended by way of our spatial intuition. Yet, despite his very traditional view of geometry, Frege was perhaps the most perceptive and precise critic of Hilbert’s new approach to axiomatic. Frege’s criticisms began when he engaged Hilbert in correspondence after reading the latter’s Foundations of Geometry. Frege began by harshly criticizing Hilbert’s use of terms like ‘axiom’ and ‘definition’. Given Frege’s somewhat condescending tone, it is unsurprising that Hilbert’s response was lukewarm, nor is it surprising that the correspondence was a brief one. Nevertheless, in the sequence of letters (and in the subsequent exchange of papers between Frege and Hilbert’s defender, Alwin Korselt) there are quite a number of thorny conceptual issues which have yet to be satisfactorily sorted out.

Subsequent commentators have largely sided with Hilbert and tend to read Frege’s attack as the result of an outmoded conception of geometry which forbid any real connection between his criticisms and the actual content of Hilbert’s epoch-making work. To put it glibly, Frege’s criticisms of Hilbert have mostly been dismissed as the curmudgeonly musings of a defender of the old guard, the dying gasp of an ancient and obsolete point of view. But, as Paul Rusnock has convincingly argued, and as I will argue myself, there is no conceptual connection between Frege’s rejection of Hilbert and his own views on geometry. Frege’s criticisms are almost entirely methodological, and are more directly connected to his views on the nature of certain basic logical concepts. (Moreoever, we can find in Frege’s writings on arithmetic the elements of a criticism of his own Kantian views of geometry). His understanding of the nature of axioms and definitions, for instance, was well ahead of its time and, as I will argue, his views (or concerns related to his views) only began to be taken seriously by Hilbert and others within the mathematical community during the great foundational crises in mathematics of the 1920s.

After having dismissed the connection between Frege’s views on geometry and his views on axiomatics, I discuss the content of his criticisms of Hilbert in detail. In order to understand the correspondence, I think we require some prior grounding in Frege’s own work, which centered around the clarification and rigorization of basic elements of mathematics as well as logic. In order to provide this grounding, I begin another brief historical portrait which should, I hope, help to show the mathematical and philosophical context in which Frege’s work occurred. His work in mathematics and logic can be seen, at least partially, as a continuation of the project of rigorization and conceptual clarification begun much earlier in the nineteenth century by the Bohemian philosopher-mathematician Bernard Bolzano and developed later by the German mathematician Karl Weierstrass and his circle of students. Bolzano and Weierstrass (among others) were concerned to place the branch of mathematics known as analysis on a firmer foundation and to achieve this goal they presented far more rigorous proofs and definitions than had hitherto been required by most mathematicians. Frege took this approach and brought it to bear on some of the most fundamental concepts of mathematics, e.g., ‘number’ and ‘function’ in an attempt motivated by the desire to show that mathematical concepts were specifically reducible to logical concepts, that mathematics was ultimately a special branch of logic. He also made huge advances in logic, both in the technical symbolic apparatus and in its conceptual underpinnings, some of which are relevant to my subsequent discussion and will accordingly be highlighted.

Frege’s criticisms of Hilbert are continuous with his work on the logical foundations of mathematics and with his insistence on very strict standards of logical rigor within mathematics. By the time of the correspondence, Frege’s work for a decade and more had centered around the painstaking reconstruction of the most basic elements of mathematics, as well as an intensely inventive approach to the rigorization of logical and mathematical proof procedures. It is not suprising, then, that his reaction to Hilbert’s fast and loose employment of terms like ‘definition’ and ‘axiom’, concepts which Frege had spent years thinking very seriously about, would have seemed irritating at best. But Hilbert’s concerns were different than Frege’s, as were the concerns of the majority of the mathematical world (which is clear from a cursory glance at the virulent polemics Frege deployed against the positions offered by the mathematical community), and Frege’s criticisms of Hilbert were met with eventual silence on Hilbert’s part, though Frege continued to respond to Korselt.

Frege’s chief criticisms of Hilbert’s work have to do with how Hilbert himself understood the nature of his results. Hilbert’s view was that his consistency and independence proofs were directly applicable to Euclidean geometry. But for Frege this was by no means obvious. For Frege functions and objects had a determinate ontological status (which I will discuss at great length later in the dissertation). So, when Hilbert moves from the uninterpreted system to its various models, the proofs resulting from that move are either about the abstract, uninterpreted structure itself, or else about the concrete instantiations given via ‘intrepretations’. By Frege’s lights, it is not at all clear how we can make sense of the sorts of claims that Hilbert makes, particularly regarding his independence proofs.

In order to better understand what it is that Hilbert might mean, Frege begins by trying to reconstruct what it is that his axioms are supposed to do. It seems clear that they are meant to define the primitive terms of some system. This is acceptable, as far as it goes, though still radically different from Frege’s own conception of axioms as primitive, intuitively grasped truths. But, even after accepting the axioms-as-definition view, Frege has trouble understanding what Hilbert is up to. First of all, if the axioms are to act as definitions, they can only define the total system of concepts and not the individual concepts in isolation. This is so because Hilbert employs primitive terms to define other primitive terms in what appears initially to be a circular manner. For example, ‘line’ will be defined by reference to ‘point’, whereas ‘point’ will subsequently be defined by reference to ‘line’. Hilbert himself admitted as much and suggested that the whole collection of axioms (or at least of circumscribed groups of axioms) is to be seen as a kind of contextual definition of a system of primitive concepts. So we can talk only of the system of concepts which is defined by any group or subgroup of the axioms and not the individual, isolated concepts themselves. Though it is obviously not to his tastes, Frege is willing to allow this form of definition to Hilbert, and even offers his own account of how this might be done rigorously. But, still, problems remain.

In addition to the collective nature of Hilbert’s ‘defining axioms’, Frege also has problems with Hilbert’s independence and consistency proofs, which were at the heart of his success. In order for Hilbert’s independence proofs to go through, we have to treat sub-groups of the axioms as if they defined the same concepts as the complete group of axioms, otherwise the ‘independence’ of one axiom from the partial group really only amounts to the internal consistency of the concepts described by that partial group, and says nothing about the larger, complete group about which we are supposed to have proved something. Because Hilbert’s definitional axioms cannot define any of the primitive concepts in isolation, any alteration in the axioms (e.g., the removal of one or the addition of another) necessarily alters the actual primitive conceptual system at issue. In other words, the alteration of the axiom set alters the very content of the discussion, so it is not at all apparent how (or if) we can speak of independence when the procedure used to prove independence opens with one system of concepts and closes with another. Similar arguments apply to the procedures required for Hilbert’s consistency proofs.

Whatever Hilbert’s axioms are capable of doing, they are not, according to Frege, capable of showing the consistency or independence of Euclidean geometry, nor are they capable of providing isolated definitions of the primitive terms of Euclidean geometry. Even stronger, it is not clear that they can prove the consistency of a set of axioms or the independence of an axiom from such a set for any system. Despite his incisive critical remarks regarding the nature of Hilbert’s view of definition and axioms, Frege does not give up entirely on Hilbert’s project. Instead he offers him the benefit of his finely honed logical vocabulary in order to show Hilbert what it is he himself has done. The last section of the chapter is dedicated to a detailed account of Frege’s proposed alterations.

In the next chapter I will argue that Hilbert only began to take Frege’s criticisms seriously much later in his career, during his ‘formalist’ period. In that chapter I will attempt to draw out the crucial dichotomy between Frege’s ontologically grounded, object-oriented approach, and Hilbert’s (initially) radically relational vision of the nature of mathematics.


Chapter Two: From Set Theory to Structuralism: Skolem’s Paradox and Benacerraf’s Dilemma

Basic aim and outline of the chapter:

The primary goal of this chapter is to connect Frege’s criticisms of Hilbert’s early axiomatic approach to Hilbert’s later formalism and also to the widespread practice of set-theoretic foundationalism within mathematics in the first half of the twentieth century. And, following this, to argue that structuralism as a position within the philosophy of mathematics arose (in large part) in response to problems with set-theoretic foundationalism pointed out by Paul Benacerraf in his 1965 paper ‘What Numbers Could Not Be’. Pursuant to this goal, I begin with a discussion of Hilbert’s formalist position and argue that it shows evidence of having taken Frege’s criticisms (or at least the general drift of those criticisms) more seriously. From here I discuss some of the conceptual aspects of the development of set-theory as a foundation for mathematics, and argue that there is a close similarity between the set-theoretic approach to axiomatics and the tempered formalism Hilbert developed in response to Frege. After showing this similarity and establishing some of the details of set-theoretic foundationalism, I move on to a few criticisms and problems with this approach, which echo in more specific forms the criticisms of Frege. Of particular interest here is Thoralf Skolem’s so-called ‘paradox’, evidence of which was first published in 1922. I will argue later in the chapter that the uproar which this ‘paradox’ (actually an interesting theorem) caused in the 1920s within mathematics and logic was mirrored much later within the philosophy of mathematics when Paul Benacerraf posed a more directly philosophical dilemma for set-theoretic reductionism. The final portion of the chapter outlines Benacerraf’s own suggestion for a ‘structuralist’ position within the philosophy of mathematics, a suggestion which will segue into the third chapter dealing with a few of the more interesting varieties of contemporary mathematical structuralism, and the issues which separate them.


Summary of the chapter:

After having discussed Frege’s suggested remedies for Hilbert’s early axiomatic approach, I move on in this chapter to Hilbert’s later, more self-consciously philosophical position, formalism.

With the increasing popularity of reductions of more complex branches of mathematics to set theory, the appearance (beginning at the dawn of the twentieth century and continuing into the 1920s) of the infamous set theoretic paradoxes had a startling effect on the mathematical community. While the new attitudes toward geometry had encouraged wildly new developments in that sub-discipline and elsewhere, the ‘truth’ (however this was conceived) of the old theorems of Euclidean geometry was never called into question. But with the set theoretic paradoxes, and the nascent belief that mathematics was ‘really’ about the universe of sets, the certainty that had pervaded mathematics for over two millennia began to wane. In response to this crisis (or cluster of crises), many gifted mathematicians attempted to shore up their science by clarifying its nature and its foundations. Of particular interest for my discussion here is the debate between formalism (championed most notably by Hilbert) and the intuitionism begun by L. E. J. Brouwer and developed considerably by Hermann Weyl, Arendt Heyting, and others.

After a brief discussion of the collapse in certainty which provides the stage for the great debates of the 1920s, I turn directly to a discussion of Hilbert’s formalism. At the most basic level, Hilbert’s position is that mathematics can be grounded in a very simple system of intuitively acceptable stroke symbols and certain transformation rules. Now, much of the reasoning employed within the more interesting areas of mathematics goes beyond the limits of the stroke symbol system alone, it requires principles like transfinite induction (for instance) which have no analogue within the system of stroke symbols. Hilbert calls the mathematics we can achieve by means of the stroke symbol system alone ‘contentual’ mathematics, while everything else is non-contentual. This picture of the foundations of mathematics is, at first glance, starkly different from Hilbert’s early attempt to maintain a radically relational picture, in which reference to a fixed ontological ground (like that implied by the formalist’s stroke symbol system) was absent. But, when we look at the relationship between the non-contentual (but ‘interesting’) portions of mathematics and contentual mathematics, we can immediately see that Hilbert still wishes to maintain large parts of the relational character he gave such importance to in his earlier work.

Here the debate between Hilbert and Brouwer (and the other intuitionists/constructivists) is helpful, and so I take a brief detour into Brouwer’s views on mathematics. Brouwer, like Hilbert, was very concerned with the appearance of antinomies within mathematics and the effect these antimonies had on mathematical certainty. Brouwer, again like Hilbert, began by developing a form of mathematics which was epistemologically tractable and whose theorems would be in a certain sense ‘surveyable’. For Brouwer this entailed a large-scale rejection of considerable portions of classical mathematics due to their reliance on intuitively unacceptable modes of reasoning (this list, for Brouwer, notably included any reasoning employing the principle of the excluded middle). It is interesting that in spirit, at least, Hilbert’s contentual mathematics and Brouwer’s intuitively acceptable mathematics are very similar. The key difference between Hilbert and Brouwer, a difference which led to a sometimes bitter personal feud, was to be found in Hilbert’s view of the connection between contentual and ostensibly non-contentual mathematics.

For, unlike the intuitionists and other radical constructivists who dismissed large portions of mathematics as meaningless, Hilbert was interested in salvaging the bulk of mathematics while avoiding the disastrous consequences of a free-wheeling set theory which seemed to have called the whole edifice into question. His solution was to accept any foray into the non-contentual realm as long as it could eventually be re-connected, at the end of a chain of non-contentual reasoning, back to the contentual world of stroke-symbols. In other words, if the theorems we prove using ‘meaningless’ methods of reasoning can be shown to remain theorems within the stroke symbol system, then those theorems remain acceptable. The fact that we had to make a foray into the strange and technically meaningless realm of transfinite reasoning does not affect the fact that the results remain valid for the limited realm of contentual mathematics. We might think of the non-contentual portions of mathematical reasoning that result in contentual theorems as loops connected to a straight line. We might simply cut the loops and leave their end points on the straight lines; the loops themselves are dispensable and their removal leaves only acceptable, meaningful mathematics.

The specifics of Hilbert’s formalism and its divergences from intuitionism are of course interesting, but more interesting for my purposes is the shift from his earlier radically relational position to a relational structure that is ultimately grounded in an ontologically fixed system. In his earlier work, Hilbert eschewed Frege’s demands for a fixed referential system, and ignored also his criticisms that the independence and consistency proofs simply could not do what they were said to do without this kind of grounding in a fixed system. By the time he developed his formalist position, however, I argue that Hilbert had begun to take the problems pointed out in Frege’s critique far more seriously (by way of Russell and Whitehead’s Principia Mathematica and his problems with his own earlier attempts to get rid of the relativity of his consistency proof of geometry through an eventual grounding in the consistency of arithmetic). His turn to the problems pointed out by Frege resulted in a new position (his formalism) which resembles in a loose sense the very position Frege himself had outlined as a remedy much earlier. Hilbert in the meantime had accepted the relativity of his consistency proofs but began (beginning about 1917) to seek a deeper, absolute consistency proof which would ground the entire relational structure, proving once and for all what he had had previously to defer.

Hilbert’s type of view (which combined the relational quality of ‘higher level’ mathematics with a belief in the ultimate reducibility of the relational structures to some basic, fixed system) became popular throughout the 20s and 30s in various guises. Of particular importance for my purposes in this chapter is the increasing popularity of set-theoretic reductionism which was developed throughout the first third of the twentieth century and which remains popular in various forms up to the present day. The foundational, set theoretic view combines the sort of axiomatic approach developed by Hilbert in his geometrical work with an ontologically fixed, absolute universe (much as Hilbert attempted to do with his formalist work). Just as Hilbert’s stroke symbols were supposed to be the anchor that guaranteed the certainty of the more aerie heights of abstract mathematics, so, too, sets were thought to be the intuitively acceptable objects which could provide a ground for things like consistency proofs in more specific mathematical areas. The problems of set theory were of course well-known at this time but various systems were proposed to remedy the situation, and slightly limited axiomatizations of set theory seemed to avoid the complications at least to the satisfaction of mathematicians not overly concerned with the philosophical motivations upon which they rested.

After having discussed Hilbert’s formalism and its analogue in set theoretic foundationalism, I turn at this point in the chapter to problems facing those sorts of views. Two instances are of particular note for my purposes, namely Skolem’s so-called paradox and Benacerraf’s dilemma. These two problems strike directly at the very conception of set theory as a fixed point which grounds the vast relational edifice of mathematics. In 1922 the Norwegian mathematician and logician Thoralf Skolem delivered a lecture (subsequently published) which illustrated quite precisely the relativity (or ‘relationality’) of set theory itself.

At this point in the chapter I discuss the details of Skolem’s so-called paradox as well as its reception, particularly its reception by set theorists like Ernst Zermelo who were busy constructing the apparatus required to treat set theory in the foundational, absolute manner discussed above. When Skolem illustrated the relational character of set theory, he undermined (I argue) the attempt to stop the regress of relational structures in an absolute universe. What Skolem’s ‘paradox’ shows is that the universe of sets itself is a relational structure, and hence the regress does not stop, or at least not in the satisfactory manner envisioned by Hilbert’s demand for an absolute consistency proof. Perhaps because of the major conceptual problem that Skolem’s work posed for the widely popular practice of set-theoretic reductionism, its meaning was not immediately apparent, even less was it accepted and understood.

Following my discussion of Skolem’s paradox and its reception, I skip ahead, historically, to 1965 and the publication of Paul Benacerraf’s paper “What Numbers Could Not Be”. If Skolem’s 1922 paper brought the relativity of set theoretic notions to the attention of the mathematical world, then Benacerraf’s paper did much the same for the philosophy of mathematics.

I turn, then, to a discussion of the content and implications of Benacerraf’s paper. The paper constructs a thought experiment in which two young girls are taught alternative set-theoretic conceptions of the natural number sequence. The underlying set theory remains the same, but the way in which the natural numbers are reduced to sets differs. One girl learns the von Neumann ordinals, while the other employs Zermelo ordinals. Now, the dilemma that Benacerraf poses is this: if the natural numbers just are sets (as many forms of reductionism hold), then which sets precisely are they? If the natural numbers ‘are’ some sets, there should be a determinate answer to the question as to which set the number 2 is, for example. Benacerraf goes on to argue what any set theorist would accept, namely that there is no principled reason for employing one or the other reduction: both adequately fulfill the demands of number theory and both are equally well-grounded in the same sort of axiomatic set theory. None of this was terribly novel at the time, and it was widely accepted that a variety of different reductions were possible. The important part of Benacerraf’s paper (for my purposes) is that he points out the strange disconnect between the belief in set theory as the ground for mathematics and the belief that multiple different methods of reduction are possible. Equally important are the makings of a positive philosophical position which Benacerraf briefly develops at the end of the paper. There he argues that the numbers aren’t any ‘objects’ at all, hence it makes no sense to ask which sets the numbers are. Rather, the numbers should be construed as the abstract elements of an abstract structure, i.e., as elements which possess no qualities other than those which are determined by their position within that structure. Benacerraf’s dilemma leads him to a structuralist position within the philosophy of mathematics, and this position begins as a radically relational account of the nature of mathematics that is similar to Hilbert’s early work. In the following chapter I begin my discussion of structuralism within the philosophy of mathematics in earnest.

Hamburger Clam

the luxury of wishing you were dead instead of being it

the secret little things for secret little people

only alive because I'm not dead

they're gonna be pissed if they lose

Group 1 Group 2

What I have and do not have.



Groupoid.

Adjoint functor, from the empty set to another empty set.

"What am I doing?"

Doing the same shit, the same ways, the same results.

God is just sitting outside of everything surveying it forever.

What an asshole.

(It's just a picture to Him, who else buys art but assholes).

Even if you see everything, you don't see everything.


"He looks like a fucking lizard"

Leadin' them lives what I want to have stole.

"Yeah, I'm fucking sure"





"It can wait, it can wait."

You sure?

"Celebrate"


Things I don't deserve and things I do deserve.


Failings aren't complete, partial things get rewards.

"Me? I get drunk. Alright?"

Stupid face, I got a stupid face. Too hung up on things to see anything but things.

"A dozen different kinds of revenge"

I want to stack the world against you.

Whatever whatever, giving and not taking.

But taking too much, too.

Everyone loses.





Is it an insane apparition that it will be me in the proper place some day?

(It is, stay in the ditch).






Why do I believe my heart anymore?


Why did I ever?


Such perfect harmony, though, with this beautiful wide world of ours.

(Ours, yes)

Kill everything, torch the bodies

I'm not like her, even a little bit.

Nothing ever fucking ends for me.

Except when I am like her, like today, shallow and cruel.

The pep talks
The barely awake and still screamings
The dance routine derailed by a broken tapedecks.

The melancholia isn't charming or cute or interesting anymore, new tactics please.

Pick it up pick it up

Mute

talking never did me any good

living like epoxy, gluing minutes together

big bag of bagels

what is it with me and alliteration

its the basis of all good




imagine if the word onomatopoeia was itself an onomatopoeia, that would be fucking so complicated to imagine how to make that work

fine feathered friends


i could just say that all day long and have a smile on my face

fuckin' A

Punctuations

, .

Queries for the Most High

Loser wins competition, locals dispute results.


Loser fined in foiled judge-bribing scandal, locals content.

"So why can't you forgive me?"

Combinatorics (n+2+3)

Back to it. Alright.

The aim of my dissertation is to provide a helpful ontological explanation of the meaning of the term 'structure' as it is used within mathematics.

The first chapter involves a brief historical overview of geometry with particular attention to the developments leading up to the publication in 1899 of David Hilbert's 'Foundations of Geometry'. After this historical introduction, I turn to a discussion of the correspondence between Gottlob Frege and Hilbert regarding the way in which the latter employed the terms 'axioms' and 'definitions'. The point of this discussion is to illustrate that Hilbert's approach to the foundations of mathematics (which became almost universally accepted in the 20th century) is susceptible to Frege's criticisms, which imply a kind of conceptual confusion between the radically relational view of mathematics that Hilbert seems to want to offer and the actually limited 'absolute' sense in which his work does in fact operate. This basic confusion, [between what I will call the 'object-oriented' approach (of Frege and, confusedly, of Hilbert as well) and the 'structure-oriented' or 'relational' approach desired by Hilbert] will form the central conceptual focus of the remainder of the thesis.

In the second chapter I again return to some historical material, with a discussion of the successes of a Hilbert-style relational axiomatic approach and it's subsequent 'anchoring' in the reductive, foundational view of set theory championed by many mathematicians and philosophers in first half of the century. Of particular importance, given my earlier discussion of the Frege-Hilbert correspondence, will be Hilbert's own later, formalist views. I will argue that Hilbert's formalist position takes into account Frege's criticisms but that, even with this alteration, the combination of a 'relational' view with an 'object-oriented' view is subject to severe ontological problems. In particular, I highlight two instances in the history of mathematics and the philosophy of mathematics where this becomes evident.

The first of these, Skolem's 'Paradox' was first brought to the attention of the public in 1922. It's reception is remarkable, Ernst Zermelo initially believed it to be a hoax. The intricacies of the so-called paradox (actually a very interesting theorem and no paradox at all) are of course important, the main reason I draw attention to this incident, however, is to illustrate further the tension between a structural view of the nature of mathematics and an absolute, object oriented view. Many mathematicians at the time had come to see mathematics as ultimately grounded in (i.e., 'about') the universe of sets. This universe was considered to be a collection of objects of some kind (there are various dispensations explaining exactly which kind). What Skolem's paradox did was to call in to question the sense of the 'reduction' of mathematics to one particular universe, or, perhaps better, he called into question the 'obviousness' of the way in which the reduction was achieved by showing that there were, by set theory's own lights, very strange sorts of models which achieved the reduction in a way that no one found intuitively plausible. Though Skolem's Paradox was thoroughly discussed, relatively few mathematicians or logicians took a purely 'structural' approach in response to it. Nevertheless, as I will show, an increasingly structural approach in mathematics arose, largely from the more algebraically oriented sub-disciplines of mathematics (group theory, algebraic topology, and eventually category theory).

This brings us to the second instance in which the tension between the object oriented view and the structural view becomes apparent. Namely, Paul Benacerraf's paper "What Numbers Could Not Be", published in 1965. There Benacerraf poses a problem for set theoretic foundationalism which is similar in spirit, if not in precise content, to the problems posed by Skolem's Paradox. He points out the widely accepted fact that there are many different possible reductions of, e.g., the natural numbers to the universe of sets. We have, for instance the von Neumann ordinals and the Zermelo ordinals, both of which can fulfill the requisite number theoretic axioms. The problem posed by this, for Benacerraf, is that the acceptability of multiple different reductions to sets calls into question the object-based view of the reduction in the first place. If the numbers just are sets, but we cannot decide which sets, EXACTLY, they are, then what sense does it make to say that the numbers 'are' sets at all?

After having posed this question, Benacerraf suggests his own answer: numbers aren't really sets at all, numbers aren't any particular mathematical object but are positions in mathematical structures, which possess only structural or relational properties. His view, which he states somewhat loosely and tentatively, suggests a movement away from the absolute, referentially fixed conception of set theory as a foundation for mathematics and towards a relative or relational view somewhat akin to Hilbert's early approach to geometrical axiomatics. As noted, however, similar concerns about the referentially fixed foundational view of mathematics had arisen within mathematics itself much earlier (with, e.g., the publication of Skolem's Paradox), and Frege had also pointed out several pressing problems related to Hilbert's view. Nevertheless, it is Benacerraf's paper (combined with the widespread mathematical influence of Bourbaki and the rise of more obviously structural mathematical sub-disciplines like category theory) that first brought the structural approach into the philosophy of mathematics in a serious way. In the wake of Benacerraf's dilemma, several structuralist positions within the philosophy of mathematics have arisen. A debate of particular importance for the later portions of my dissertation has been between realist structuralists, who take structures to possess some sort of ontological importance, and nominalist structuralists, who view structures as a mere facon de parler. This debate obviously has much earlier roots in the philosophical positions of Plato, Aristotle, and many others regarding the nature of mathematical objects, but it begins to take on a highly specialized form within the philosophy of mathematics in the second half of the 20th century.

After establishing the historical links between Hilbert\s formalism, reductionist set-theoretic foundationalism, and the confusion between object- and structure-oriented views, the third chapter turns directly to more recent structuralist approaches. While there are quite a number of different approaches in the philosophy of mathematics which can be called 'structuralist', I am interested (initially at least) in two main varieties; namely, realist structuralism and nominalist structuralism. As my eventual aim in this dissertation is to establish a satisfactory view of 'structure' as a basic ontological category (a view in which 'structures' are accorded some kind of real ontological status), I begin my discussion of structuralist positions with opposed views in order to undermine both their motivations and their specific claims. There are several disparate 'nominalist' structuralisms, but to my mind the two most important advocates have been Geoffrey Hellman, with his modal, eliminative view (the non structures view) and Charles Chihara's constructive approach. While the specifics of both of these programmes are very interesting, my interest is to undermine the motivation for a nominalist view of the nature of mathematics. I attempt to do so by arguing against both the usefulness of a nominalist reduction (for mathematics and philosophy both), and also by suggesting that neither of these positions achieves a truly 'nominalist' view of mathematics, even by their own lights, precisely insofar as they rely on certain kinds of abstract entities that they forbid their realist counterparts access to.

After having seriously reduced the motivation for nominalism within the philosophy of mathematics, I turn to a discussion of it's broad alternative: realism.

In the last 25 years or so the two major advocates of structural realism have been Michael Resnik and Stewart Shapiro. Resnik views structures as 'patterns' which can be exemplified within or instantiated by particular systems of objects. Shapiro's view takes structures to be sui generis universals, kinds of abstract objects which exist independently of any possible or actual realization in a given system of objects. I explore these two realist positions in some detail in order to provide a sense of the problems facing such views more generally. Particularly important for my interests later in the dissertation are problems centering around notions of mathematical existence, ontological commitment, and the conceptual differentiation between mathematical structures (or patterns) and mathematical 'objects'.

The result of my dissection of the debate between realist and nominalist forms of structuralism should be the sense that the same ontological problems ultimately face both sorts of views, and also that neither strand has developed sufficiently clear ontological concepts capable of filling in the well-known gaps. At this point in the chapter I turn to a third form of structuralism which arose largely as an internal development within mathematics itself, namely, category-theoretic structuralism. Since it's early days (beginning with a famous 1945 paper on natural transformations by Saunders Mac Lane and Samuel Eilenberg) has been characterized as a possible alternative 'foundation' for mathematics which is conceptually and technically quite different from set theory. Just how the foundational nature of category theory is to be understood however is a question which has garnered much debate, beginning with F. William Lawvere's development of a category-theoretic treatment of axiomatic set theory.

Category theoretic structuralism within mathematics itself is neither definitively realist nor definitively nominalist in most of its incarnations, but is often treated simply as a technique to be used appropriately within a wide variety of mathematical contexts. My aim, in this and the next chapter, is to show that an appropriate ontological grasp of the concepts at work within the category-theoretic, foundational approach can help us avoid the confusion between object- and structure-oriented approaches that has plagued foundational views since at least Hilbert's 1899 work (and likely much earlier).

After having outlined my reasons for rejecting nominalism as a viable variety of mathematical structuralism in the previous chapter, I turn in the fourth chapter to my own positive project. This project consists, in large part, in combining a category-theoretic form of foundationalism within mathematics with my own ontological conception of structures. In this fourth chapter I focus on the problems facing a realist approach to category theory, as well as upon the problems we must face if we want to avoid the confusion between a referentially fixed, object-oriented approach and a relational, structure-oriented approach.

I discuss several varying conceptions of the nature of category theoretic structuralism. Of particular interest is an early (1977) paper by Solomon Feferman in which he argues that category theory, like set theory, requires a pre-theoretic grounding in concepts like 'collection' and thus does not offer us anything essentially new (in this regard). In other words, Feferman believes that the basic concepts of category theory, too, lead us to a non-structural ground and that, therefore, category theory does not offer an essentially 'structural' view any more than does set theory. By contrast, several later views either dismiss Feferman's view that category theory is or could be foundational for mathematics in the same way that set theory seems to have been, or they argue that category theory shows us that we can treat every notion as structural (in which case 'objects' themselves simply become structures when viewed from a particular mathematical position).

Taking Feferman's interesting paper as a foil, I argue both points. First I argue that category theory should not be treated as foundational for mathematics in the same sense that set theory has been (i.e., we should not see mathematical disciplines of greater complexity and specificity as supervening on the more fundamental mathematical/ontological universe in which categories are the only object). In this direction I follow a suggestion (made distinct by Elaine Landry) that we treat category theory as one, perhaps the best, language in which mathematics can be formulated. The benefit of this understanding is that the 'strictly mathematical' use of category theory is partially explained: category theory on this level simply provides us with an extraordinarily versatile and fruitful tool which can be applied in almost any mathematical context, as Mac Lane, Lawvere, and others have shown.

But, despite (or in addition to) this view of category theory as a framework for mathematics, I also want to combat Feferman's suggestion that category theory must ultimately rely upon non-structural conceptual elements in precisely the same way that set theory requires. What I suggest in the second half of this chapter is that the Feferman-type view that all foundational approaches require a ground in non-structural elements results from the 100 year old confusion between object-oriented views and structure-oriented views. At this point I begin my ontological focus on the twin concepts of 'object' and 'structure' and argue that the historical confusion has arisen (and continues) because mathematicians and philosophers, both, have had a conception of 'structure' which is insufficiently different, on an ontological level, than that a common notion of 'object'.

The sense derived from this chapter should be that a category theoretic, realist approach to structuralism in the philosophy of mathematics is possible, and also that it has run headlong into an ontological debate that has resisted clarification due to an unnecessary confusion between the ontological differences between structures and objects. In the next chapter I aim to tackle these ontological problems more directly.

In the fifth chapter I turn away from the specificities of the philosophy of mathematics and enter the more general forum of (formal) ontology.

I begin my discussion by returning to the work of Gottlob Frege, this time, however, to his fundamental distinction between concepts [or functions] and objects. Frege's distinction is interesting for a number of reasons. First, it is interesting insofar as it highlights the limits of rigorous systematization that Frege (and Hilbert) aspired to achieve.

The concepts of 'concept' and 'object' are for Frege indefinable (given his understanding of the nature of definition), but can become explicated though appropriate non-systematic, pre-scientific discussions.

Frege's explications seem to have the character of metaphors: for him objects are self-subsistent entities, fully saturated, and complete, requiring no supplementation. Concepts, by contrast, require supplementation, are unsaturated, cannot stand alone, etc.

In order to further explicate Frege's use of these terms, I turn to the famous view held by Frege that "the concept 'horse' is not a concept"

There does seem to be room within a Frege-style framework for a 'structural' view, if we treat concepts as structures. But it seems clear that for Frege this would have been an unsatisfying or plainly false view to take. For him, mathematics and logic, both, were grounded in the total system of thoughts, and this system was composed of objects (i.e., thoughts) which were the result of combinations of concepts and objects. The idea of a purely conceptual, i.e., relational, science made little sense to Frege: it would have no content, and could make no claims. We would have to turn the concepts into objects (by way of the 'concept horse' maneuver) in order to say anything meaningful, and in doing so the science would lose its structural/relational quality.

In order, then, to supplement Frege's view that science is non-structural (and hence rooted in objects), I turn to the more developed remarks of Alexius Meinong regarding the nature of objects (as elaborated in his 'Theory of Objects').

One of the most intriguing views of Meinong's work can be found in Gustav Bergmann's book 'Realism: A Critique of Meinong and Brentano'. There Bergmann characterizes Meinong's object theory as an attempt to treat relations purely on the basis of objects. Bergmann argues that this leads Meinong into denser and denser ontological problems, with the ultimate result that he broadens the concept of an object (improving our understanding of that notion considerably), while nevertheless failing in his ultimate goal. A proper ontological grounding of relations in a purely object-oriented view like Meinong's is, according to Bergmann, essentially impossible.

As will become clear, in order to pursue Frege's view of mathematics as rooted in a system of objects while also maintaining the importantly relational qualities of mathematics highlight by Hilbert and (later) by the category-theoretic structuralism I endorse, we are lead down Meinong's rich ontological path. Ultimately, though, even the richest object-based ontology comes up short in its ability to explain the radical relationality of mathematics as it is practiced.

Because Bergmann views Meinong's object theory to be the furthest end of object-based trajectory, he argues that its inability to adequately capture the nature of relations signals the sterility of any object-oriented approach. Accordingly, he embarks (in 'Realism' and many subsequent publications) to develop his own fundamental ontological picture in which a place is made for a fundamental, non-object category of ontological entity (he variously uses the expression 'nexus' and 'fundamental tie' to capture this notion).

I begin with the basic project of Bergmann but diverge heavily in specifics when I attempt to construct my own positive ontological view of the nature of structures. This view will be developed partly with the demands of category theory (understood as a framework or 'language' for mathematics) in mind, and partly with the Bergmann-inspired resistance to non-structural ontologies like those of Frege and Meinong in mind.