Re: [peirce-l] Proemial: On The Origin Of Experience

2012-03-13 Thread John Collier
I agree with that, Steven. We forget how many bad paths Einstein went down 
before he relied on a friend for key input when working on General Relativity. 
It's all in his notebooks from the time.
 
John


 
 
Professor John Collier  
Philosophy, University of KwaZulu-Natal
Durban 4041 South Africa
T: +27 (31) 260 3248 / 260 2292
F: +27 (31) 260 3031
email: colli...@ukzn.ac.za On 2012/03/13 at 08:38 AM, in message 
5a506354-b312-4ebf-b5c9-7ee33401a...@iase.us, Steven Ericsson-Zenith 
ste...@iase.us wrote:


Thanks John. 

If the right question is asked and understood, then the answer is readily 
apparent if the data that confirms or denies it is accessible. In effect, the 
answers are all out there, we need only craft the right question. Scientific 
interpretation of data is but a process of question refinement and this can be 
generalized to all forms of interpretation. Contrary to the common idea that 
interpretation is some posterior act.

When we have the answer, we tend to forget the paths that either failed or were 
incomplete on our way to it.

With respect,
Steven

--
Dr. Steven Ericsson-Zenith
Institute for Advanced Science  Engineering
http://iase.info







On Mar 12, 2012, at 1:58 AM, John Collier wrote:

 
 
  
  
 Professor John Collier  
 Philosophy, University of KwaZulu-Natal
 Durban 4041 South Africa
 T: +27 (31) 260 3248 / 260 2292
 F: +27 (31) 260 3031
 email: colli...@ukzn.ac.za On 2012/03/06 at 11:03 PM, in message 
 4a39e6c5-939f-49ba-bc6b-8af976028...@iase.us, Steven Ericsson-Zenith 
 ste...@iase.us wrote:
 
 I'm not sure I would say that the Mars lander computational analysis of data 
 is interpretation. It seems to me to be a further representation, although 
 one filtered by a machine imbued with our intelligence. Interpretation would 
 be the thing done by scientists on earth.
 
 As a former planetary scientist, I would agree in general with this, but I 
 also experienced new data that pretty much implied directly (along with other 
 well-known principles) that lunar differentiation had occurred. (Even then, 
 scientists had to interpret the results, but they were clear as crystal 
 relative to the question.) I relied on much less direct data (gravity 
 evidence and some general principles of physics and geochemistry) to argue 
 for the same conclusion. My potential paper was scooped, and I hadn't even 
 graduated yet. Both Harvard and MIT people in the field found my paper very 
 interesting but lost complete interest when I was retrospectively scooped by 
 firmer evidence. The moral is that nothing in science beats direct evidence, 
 even the most appealing hypothesis. Nonetheless, your book sound interesting.
  
 Regards,
 John
 
 Please find our Email Disclaimer here--: http://www.ukzn.ac.za/disclaimer
 


Please find our Email Disclaimer here: http://www.ukzn.ac.za/disclaimer/

-
You are receiving this message because you are subscribed to the PEIRCE-L 
listserv.  To remove yourself from this list, send a message to 
lists...@listserv.iupui.edu with the line SIGNOFF PEIRCE-L in the body of the 
message.  To post a message to the list, send it to PEIRCE-L@LISTSERV.IUPUI.EDU


Re: [peirce-l] Mathematical terminology, was, review of Moore's Peirce edition

2012-03-13 Thread Irving

Ben, Gary, Malgosia, list

It would appear from the various responses that. whereas there is a 
consensus that Peirce's theorematic/corollarial distinction has 
relatively little, if anything, to do with my theoretical/computational 
distinction or Pratt's creator and consumer distinction.


As you might recall, in my initial discussion, I indicated that I found 
Pratt's distinction to be somewhat preferable to the 
theoretical/computational, since, as we have seen in the responses, 
computational has several connotations, only one of which I initially 
had specifically in mind, of hack grinding out of [usually numerical] 
solutions to particular problems, the other generally thought of as 
those parts of mathematics taught in catch-all undergrad courses that 
frequently go by the name of Finite Mathematics and include bits and 
pieces of such areas as probability theory, matrix theory and linear 
algebra, Venn diagrams, and the like). Pratt's creator/consumer is 
closer to what I had in mind, and aligns better, and I think, more 
accurately, with the older pure (or abstract or theoretical) vs. 
applied distinction.


The attempt to determine whether, and, if so, how well, Peirce's 
theorematic/corollarial distinction correlates to the 
theoretical/computational or creator/consumer distinction(s) was not 
initially an issue for me. It was raised by Ben Udell when he asked me: 
Do you think that your theoretical - computational distinction and 
likewise Pratt's creator - consumer distinction between kinds of 
mathematics could be expressed in terms of Peirce's theorematic - 
corollarial distinction?


I attempted to reply, based upon a particular quote from Peirce. What I 
gather from the responses to that second round is that the primary 
issue with my attempted reply was that Peirce's distinction was bound 
up, not with the truth of the premises, but rather with the method in 
which theorems are arrived at. If I now understand what most of the 
responses have attempted to convey, the theorematic has to do with the 
mechanical processing of proofs, where a simple inspection of the 
argument (or proof) allows us to determine which inference rules to 
apply (and when and where) and whether doing so suffices to demonstrate 
that the theorem indeed follows from the premises; whereas the 
corollarial has to do with intuiting how, or even if, one might get 
from the premises to the desired conclusion. In that case, I would 
suggest that another way to express the theorematic/corollarial 
distinction is that they concern the two stages of creating 
mathematics; that the mathematician begins by examining the already 
established mathematics and asks what new mathematics might be


Ben Udell also introduces the issue of the presence of a lemma in a 
proof as part of the distinction between theorematic and corollarial. 
His assumption seems to be that a lemma is inserted into a proof to 
help carry it forward, but is itself not proven. But, as Malgosia has 
already noted, the lemma could itself have been obtained either 
theorematically or corollarially. In fact, most of us think of a lemma 
as a minor theorem, proven along the way and subsequently used in the 
proof of the theorem that we're after.


I do not think that any of this obviates the main point of the initial 
answer that I gave to Ben's question, that neither my 
theoretical/computational distinction nor Pratt's creator and 
consumer distinction have anything to do with Peirce's 
theorematic/corollarial distinction.


In closing, I would like to present two sets of exchanges; one very 
recent (actually today, on FOM, with due apologies to the protagonists, 
if I am violating any copyrights) between probability theorist William 
Taylor (indicated by '') and set theorist Martin Dowd (indicated by 
''), as follows:



More seriously, any freshman philosopher encounters the fact that there are
 fundamental differences between physical reality and mathematical reality.


Quite so.  And one of these is noted by Hilbert (or maybe Hardy, 
anyone help?) -



The chief difference between scientists and mathematicians is that
mathematicians have a much more direct connection to reality.



This does not entitle philosophers to characterize mathematical reality
as  fictional.



Quite so; but philosophers tend to have a powerful sense of entitlement.


the other, in Gauss's famous letter November 1, 1844 to astronomer 
Heinrich Schumacher regarding Kant's philosophy of mathematics, that: 
you see the same sort of [mathematical incompetence] in the 
contemporary philosophers Don't they make your hair stand on end 
with their definitions? ...Even with Kant himself it is often not much 
better; in my opinion his distinction between analytic and synthetic 
propositions is one of those things that either run out in a triviality 
or are false.


- Message from bud...@nyc.rr.com -
   Date: Mon, 12 Mar 2012 13:47:10 -0400
   From: Benjamin Udell 

Re: [peirce-l] Book Review: Peirce and the Threat of Nominalism

2012-03-13 Thread Catherine Legg
Michael I just read the book review from Nathan Houser you shared - it is
lucidly written over 6 pages and gives a commanding overview of Peirce's
realism. I really enjoyed reading it, thanks for posting it.

Cathy

On Fri, Mar 9, 2012 at 6:13 PM, Michael DeLaurentis
michael...@comcast.netwrote:

 If there has already been a post about this, my apologies. Book review
 just in on CSP and nominalism. 

 ** **

 Michael J DeLaurentis

 ** **

  

 ** **

 ** **

 -
 You are receiving this message because you are subscribed to the PEIRCE-L
 listserv. To remove yourself from this list, send a message to
 lists...@listserv.iupui.edu with the line SIGNOFF PEIRCE-L in the body
 of the message. To post a message to the list, send it to
 PEIRCE-L@LISTSERV.IUPUI.EDU

-
You are receiving this message because you are subscribed to the PEIRCE-L 
listserv.  To remove yourself from this list, send a message to 
lists...@listserv.iupui.edu with the line SIGNOFF PEIRCE-L in the body of the 
message.  To post a message to the list, send it to PEIRCE-L@LISTSERV.IUPUI.EDU


Re: [peirce-l] Mathematical terminology, was, review of Moore's Peirce edition

2012-03-13 Thread Eugene Halton
Dear Irving, 
A digression, from the perspective of art. You quote probability 
theorist William 
Taylor and set theorist Martin Dowd as saying: 

 The chief difference between scientists and mathematicians is that
 mathematicians have a much more direct connection to reality.

 This does not entitle philosophers to characterize mathematical reality
 as  fictional.


Yes, I can see that.

But how about a variant: 

The chief difference between scientists, mathematicians, and artists is that
artists have a much more direct connection to reality.

This does not prevent scientists and mathematicians to characterize artistic 
reality
as fictional, because it is, and yet, nevertheless, real.

This is because scientist's and mathematician's map is not the territory, yet 
the artist's art is both. 

Gene Halton



-Original Message-
From: C S Peirce discussion list [mailto:PEIRCE-L@LISTSERV.IUPUI.EDU] On Behalf 
Of Irving
Sent: Tuesday, March 13, 2012 4:34 PM
To: PEIRCE-L@LISTSERV.IUPUI.EDU
Subject: Re: [peirce-l] Mathematical terminology, was, review of Moore's Peirce 
edition

Ben, Gary, Malgosia, list

It would appear from the various responses that. whereas there is a 
consensus that Peirce's theorematic/corollarial distinction has 
relatively little, if anything, to do with my theoretical/computational 
distinction or Pratt's creator and consumer distinction.

As you might recall, in my initial discussion, I indicated that I found 
Pratt's distinction to be somewhat preferable to the 
theoretical/computational, since, as we have seen in the responses, 
computational has several connotations, only one of which I initially 
had specifically in mind, of hack grinding out of [usually numerical] 
solutions to particular problems, the other generally thought of as 
those parts of mathematics taught in catch-all undergrad courses that 
frequently go by the name of Finite Mathematics and include bits and 
pieces of such areas as probability theory, matrix theory and linear 
algebra, Venn diagrams, and the like). Pratt's creator/consumer is 
closer to what I had in mind, and aligns better, and I think, more 
accurately, with the older pure (or abstract or theoretical) vs. 
applied distinction.

The attempt to determine whether, and, if so, how well, Peirce's 
theorematic/corollarial distinction correlates to the 
theoretical/computational or creator/consumer distinction(s) was not 
initially an issue for me. It was raised by Ben Udell when he asked me: 
Do you think that your theoretical - computational distinction and 
likewise Pratt's creator - consumer distinction between kinds of 
mathematics could be expressed in terms of Peirce's theorematic - 
corollarial distinction?

I attempted to reply, based upon a particular quote from Peirce. What I 
gather from the responses to that second round is that the primary 
issue with my attempted reply was that Peirce's distinction was bound 
up, not with the truth of the premises, but rather with the method in 
which theorems are arrived at. If I now understand what most of the 
responses have attempted to convey, the theorematic has to do with the 
mechanical processing of proofs, where a simple inspection of the 
argument (or proof) allows us to determine which inference rules to 
apply (and when and where) and whether doing so suffices to demonstrate 
that the theorem indeed follows from the premises; whereas the 
corollarial has to do with intuiting how, or even if, one might get 
from the premises to the desired conclusion. In that case, I would 
suggest that another way to express the theorematic/corollarial 
distinction is that they concern the two stages of creating 
mathematics; that the mathematician begins by examining the already 
established mathematics and asks what new mathematics might be

Ben Udell also introduces the issue of the presence of a lemma in a 
proof as part of the distinction between theorematic and corollarial. 
His assumption seems to be that a lemma is inserted into a proof to 
help carry it forward, but is itself not proven. But, as Malgosia has 
already noted, the lemma could itself have been obtained either 
theorematically or corollarially. In fact, most of us think of a lemma 
as a minor theorem, proven along the way and subsequently used in the 
proof of the theorem that we're after.

I do not think that any of this obviates the main point of the initial 
answer that I gave to Ben's question, that neither my 
theoretical/computational distinction nor Pratt's creator and 
consumer distinction have anything to do with Peirce's 
theorematic/corollarial distinction.

In closing, I would like to present two sets of exchanges; one very 
recent (actually today, on FOM, with due apologies to the protagonists, 
if I am violating any copyrights) between probability theorist William 
Taylor (indicated by '') and set theorist Martin Dowd (indicated by 
''), as follows:

 More seriously, any freshman philosopher encounters the fact 

Re: [peirce-l] Mathematical terminology, was, review of Moore's Peirce edition

2012-03-13 Thread Benjamin Udell
Irving, all,

In my previous post I said that I would include the full Peirce quotes, but 
for the first Peirce quote I included only the portion included in the Commens 
Dictionary. For the full quote (CP 4.233), go here: 
http://books.google.com/books?id=3JJgOkGmnjECpg=RA1-PA193lpg=RA1-PA193dq=%22Mathematics+is+the+study+of+what+is+true+of+hypothetical+states+of+things%22

- Original Message - 
From: Benjamin Udell 
To: PEIRCE-L@LISTSERV.IUPUI.EDU 
Sent: Tuesday, March 13, 2012 6:11 PM
Subject: Re: [peirce-l] Mathematical terminology, was, review of Moore's Peirce 
edition

Irving, Gary, Malgosia, list,

Irving, I'm sorry that I gave you the impression that I think that a lemma is 
something helpful but unproven inserted into a proof. I mean a theorem placed 
in among the premisses to help prove the thesis. Its proof may be offered then 
and there, or it may be a theorem from (and already proven in) another branch 
of mathematics, to which the reader is referred. At any rate it is as Peirce 
puts it a demonstrable proposition about something outside the subject of 
inquiry. 


The idea that theorematic reasoning often involves a lemma comes not from me 
but from Peirce. Theorematic reasoning, in Peirce's view, involves 
experimentation on a diagram, which may consist in a geometrical form, an array 
of algebraic expressions, a form such as All __ is __, etc.  I don't recall 
his saying anything to suggest that theorematic reasoning is particularly 
mechanical.  I summarized Peirce's views in a paragraph in my first post on 
these questions, and I'll reproduce it, this time with the full quotes from 
Peirce. He discusses lemmas in the third quote.
Peirce held that the most important division of kinds of deductive reasoning is 
that between corollarial and theorematic. He argued that, while finally all 
deduction depends in one way or another on mental experimentation on schemata 
or diagrams,[1] still in corollarial deduction it is only necessary to imagine 
any case in which the premisses are true in order to perceive immediately that 
the conclusion holds in that case, whereas theorematic deduction is deduction 
in which it is necessary to experiment in the imagination upon the image of the 
premiss in order from the result of such experiment to make corollarial 
deductions to the truth of the conclusion.[2]  He held that corollarial 
deduction matches Aristotle's conception of direct demonstration, which 
Aristotle regarded as the only thoroughly satisfactory demonstration, while 
theorematic deduction (A) is the kind more prized by mathematicians, (B) is 
peculiar to mathematics,[1] and (C) involves in its course the introduction of 
a lemma or at least a definition uncontemplated in the thesis (the proposition 
that is to be proved); in remarkable cases that definition is of an abstraction 
that ought to be supported by a proper postulate..[3]


1 a b Peirce, C. S., from section dated 1902 by editors in the Minute Logic 
manuscript, Collected Papers v. 4, paragraph 233, quoted in part in 
Corollarial Reasoning in the Commens Dictionary of Peirce's Terms, 
2003-present, Mats Bergman and Sami Paavola, editors, University of Helsinki.: 

  How it can be that, although the reasoning is based upon the study of an 
individual schema, it is nevertheless necessary, that is, applicable, to all 
possible cases, is one of the questions we shall have to consider. Just now, I 
wish to point out that after the schema has been constructed according to the 
precept virtually contained in the thesis, the assertion of the theorem is not 
evidently true, even for the individual schema; nor will any amount of hard 
thinking of the philosophers' corollarial kind ever render it evident. Thinking 
in general terms is not enough. It is necessary that something should be DONE. 
In geometry, subsidiary lines are drawn. In algebra permissible transformations 
are made. Thereupon, the faculty of observation is called into play. Some 
relation between the parts of the schema is remarked. But would this relation 
subsist in every possible case? Mere corollarial reasoning will sometimes 
assure us of this. But, generally speaking, it may be necessary to draw 
distinct schemata to represent alternative possibilities. Theorematic reasoning 
invariably depends upon experimentation with individual schemata. We shall find 
that, in the last analysis, the same thing is true of the corollarial 
reasoning, too; even the Aristotelian demonstration why. Only in this case, 
the very words serve as schemata. Accordingly, we may say that corollarial, or 
philosophical reasoning is reasoning with words; while theorematic, or 
mathematical reasoning proper, is reasoning with specially constructed 
schemata. (' Minute Logic', CP 4.233, c. 1902)

2. Peirce, C. S., the 1902 Carnegie Application, published in The New Elements 
of Mathematics, Carolyn Eisele, editor, also transcribed by Joseph M. Ransdell, 
see From Draft A - MS L75.35-39 in 

Re: [peirce-l] Categorical Aspects of Abduction, Deduction, Induction

2012-03-13 Thread Jon Awbrey

Peircers,

I think it's true that some of the difficulties of this discussion may be due to
different concepts of predicates, or different ways of using the word 
predicate
in different applications, communities, and contexts.

If I think back to the variety of different communities of interpretation
that I've had the fortune or misfortune of passing through over the years,
I can reckon up at least this many ways of thinking about predicates:

1.  In purely syntactic contexts, a predicate is just a symbol,
a syntactic element that is subject to specified rules of
combination and transformation.

As we pass to contexts where predicate symbols are meant to have meaning,
most disciplines of interpretation will be very careful, at first, about
drawing a firm distinction between a predicate symbol and the object it
is intended to denote.  For example, in computer science, people tend
to use forms like constant name, function name, predicate name,
type name, variable name, and so on, for the names that denote
the corresponding abstract objects.

When it comes to what information a predicate name conveys,
what kind of object the predicate name denotes, or finally,
what kind of object the predicate itself is imagined to be,
we find that we still have a number of choices:

2.  Predicate = property, the intension a concept or term.
3.  Predicate = collection, the extension of a concept or term.
4.  Predicate = function from a universe domain to a boolean domain.

It doesn't really matter all that much in ordinary applications which you 
prefer,
and there is some advantage to keeping all the options open, using whichever one
appears most helpful at a given moment, just so long as you have a way of moving
consistently among the alternatives and maintaining the information each 
conveys.

Regards,

Jon

SE = Steven Ericsson-Zenith

SE: Ben and I appear to be speaking across each other and, possibly, agreeing 
fiercely.

SE: Recall that in the 1906 dialectic Peirce is drawing a distinction between 
the wider usage
of Category at the time, i.e., Aristotle's Categories considered by you 
in the dialog,
and saying that he prefers to call these Predicaments.  Having made this 
distinction he
then speaks about the indices that are his categories.

SE: As I said earlier, the index in this case does not point to the elements of 
the category
but the category itself. There is Firstness as opposed to x is a first. 
 The confusion
may be that Ben thinks I am saying that a category is some set of indices 
to its members.
That is not the case, a category stands alone and we can point to it 
(index).  Icons are
the selection mechanisms of properties of classes, not indices.

SE: Predicaments are higher order, assertions about assertions, predicates of 
predicates,
I prefer to say predicated predicates or assertions about assertions 
which is more
generally understood today.

SE: Being as careful as he is, I see no evidence to cause us to suppose that 
the categories that
Peirce attributes to himself in 1906 are different than those he identifies 
as early as 1866.

--

academia: http://independent.academia.edu/JonAwbrey
inquiry list: http://stderr.org/pipermail/inquiry/
mwb: http://www.mywikibiz.com/Directory:Jon_Awbrey
oeiswiki: http://www.oeis.org/wiki/User:Jon_Awbrey
word press blog 1: http://jonawbrey.wordpress.com/
word press blog 2: http://inquiryintoinquiry.com/

-
You are receiving this message because you are subscribed to the PEIRCE-L listserv.  To 
remove yourself from this list, send a message to lists...@listserv.iupui.edu with the 
line SIGNOFF PEIRCE-L in the body of the message.  To post a message to the 
list, send it to PEIRCE-L@LISTSERV.IUPUI.EDU


Re: [peirce-l] Deacon's incompleteness and Peirce's infinity

2012-03-13 Thread Catherine Legg
Very rich post, Gary (F), thank you! I've recently been alerted to the
importance of Deacon by Gary (R) and he is now 'on my list'.

On the interesting issue of Deacon's 'Absence' which you raise in the last
paragraph, I wonder whether the Absent is absent from Being or just the
actual world. If the latter, perhaps it is not entirely inaccessible to a
Peircean phaneroscopy fearlessly navigating the Platonic Universe.

Cheers, Cathy



On Mon, Mar 12, 2012 at 4:24 AM, Gary Fuhrman g...@gnusystems.ca wrote:

 Jon, Gary, Ben and List,

 ** **

 There's another part of the *Minute Logic* which may be related to the
 connection Jon is making between “objective logic” and “categories”. It is
 definitely related to the argument in Terrence Deacon's *Incomplete Nature
 *, which Gary R. suggested some time ago as worthy of study here. We
 haven't found a way to study it systematically, but maybe it's just as well
 to do it one post at a time. Or one thread at a time, if replies ensue.***
 *

 ** **

 The central part of Deacon's argument presents “a theory of emergent
 dynamics that shows how dynamical process can become organized around and
 with respect to possibilities not realized” (Deacon, p. 16). Depending on
 the context, he also refers to these “possibilities not realized” as
 “absential” or “ententional”. His argument is explicitly anti-nominalistic
 and acknowledges the reality of a kind of final causation in the physical
 universe (“teleodynamics”). It has a strong affinity with Peirce's argument
 for a mode of being which has its reality *in futuro*. In other words, he
 argues for the reality of Thirdness without calling it that – indeed
 without using Peirce's phaneroscopic categories at all. (Personally i doubt
 that he is familiar enough with them to use them fluently, but maybe he
 decided not to use them for some reason.)

 ** **

 “Incompleteness” is a crucial concept of what i might call Deaconian
 realism. In physical terms, it is connected with Prigogine's idea of 
 *dissipative
 structures* (including organisms) as *far from equilibrium* in a universe
 where the spontaneous tendency is *toward* equilibrium, as the Second Law
 of thermodynamics would indicate. Teleodynamic processes take
 incompleteness to a higher level of complexity, but i don't propose to go
 into that now. Instead i'll present here a Peircean parallel to Deacon's
 “incompleteness”. The connection lies in the fact that *incompleteness*is 
 etymologically – and perhaps mathematically? – equivalent to
 *infinity*.

 ** **

 First, we have this passage from Peirce's Minute Logic of 1902:

 ** **

 [[[ I doubt very much whether the Instinctive mind could ever develop into
 a Rational mind. I should expect the reverse process sooner. The Rational
 mind is the Progressive mind, and as such, by its very capacity for growth,
 seems more infantile than the Instinctive mind. Still, it would seem that
 Progressive minds must have, in some mysterious way, probably by arrested
 development, grown from Instinctive minds; and they are certainly
 enormously higher. The Deity of the Théodicée of Leibniz is as high an
 Instinctive mind as can well be imagined; but it impresses a scientific
 reader as distinctly inferior to the human mind. It reminds one of the view
 of the Greeks that Infinitude is a defect; for although Leibniz imagines
 that he is making the Divine Mind infinite, by making its knowledge Perfect
 and Complete, he fails to see that in thus refusing it the powers of
 thought and the possibility of improvement he is in fact taking away
 something far higher than knowledge. It is the human mind that is infinite.
 One of the most remarkable distinctions between the Instinctive mind of
 animals and the Rational mind of man is that animals rarely make mistakes,
 while the human mind almost invariably blunders at first, and repeatedly,
 where it is really exercised in the manner that is distinctive of it. If
 you look upon this as a defect, you ought to find an Instinctive mind
 higher than a Rational one, and probably, if you cross-examine yourself,
 you will find you do. The greatness of the human mind lies in its ability
 to discover truth notwithstanding its not having Instincts strong enough to
 exempt it from error. ]] CP 7.380 ]

 ** **

 This suggests to me that fallibility – which not even Peirce attributes to
 God – is a highly developed species of incompleteness. The connection with
 infinity, and with Thirdness, is further brought out in Peirce's Harvard
 Lecture of 1903 “On Phenomenology”:

 ** **

 [[[ The third category of which I come now to speak is precisely that
 whose reality is denied by nominalism. For although nominalism is not
 credited with any extraordinarily lofty appreciation of the powers of the
 human soul, yet it attributes to it a power of originating a kind of ideas
 the like of which Omnipotence has failed to create as real objects, and
 those general conceptions which men