> Peikoff therein talked about mathematics being atypical (epistemologically) because it was deductive...Has he modified this view since 1983?

Roger, this is the kind of thing I would have qualified in my mind with the word 'largely' or 'basically', had i heard it, assuming he realized many concepts were abstracted from (induced) from concrete instances and he was 'essentializing'.

I would definitely say that numbers are taught inductively. You show a child five blocks, tell them this is "five". You show the child five sticks, say that is "five". You repeat this with different numbers and objects till the child gets the idea of quantity.

I cannot address your question with respect to Peikoff and mathematics as a deductive field directly.

I would observe that mathematics operates as systems of essentially logical rules applied to abstract quantitities, wherein there is a realm in which mathematics is deductive. The applicability of any given set of mathematical rules to informing us about reality is entirely inductive, however. In other words, we only know whether a theory of mathematics rules can be applied to order and explain our knowledge of real entities and their actions on an inductive basis. Without informing our use of mathematics inductively, math is just a game people play for their entertainment. It becomes useful and essential only when we test its use inductively.

So, if you define mathematics very narrowly, you can say it is deductive and not inductive. If you wish to include its use for describing reality in a generous measure, then it is heavily inductive as well as deductive. Academic mathematics departments can be very oriented toward only deductive concerns, but an Applied Mathematics department, or engineers, or most scientists, must be very concerned about the inductive evaluative side of mathematics. Since I am not aware that Peikoff has told us where this boundary is with respect to his statements, I would not quibble with him on this issue. However, I am biased in the direction of thinking that math has its great value as a tool for organizing, predicting, and explaining knowledge about reality. There is a caveat here with respect to explaining reality however, since it really may only explain in the context of organizing and predicting in the context that it works well with respect to our inductive knowledge. Of course, others will say that applying math is the realm of science, but there is plenty of room for applied mathematicians and for mathematical scientists in some sort of boundary area.

Could it not be said that the concept of "quantity" itself is arrived at inductively? We arrive at the abstract concept of "quantity" by identifying a common quality in a broad class of entities– ie: all entities-- and subtracting all other qualities from observed entities. It is no different to arriving at the concept of causality by identifying the common quality in a broad class of phenomena– actions and interactions of entities-- and subtracting all other observed qualities.

Also I tend to see the basic mathematical manipulators – eg: adding, subtracting, etc.– as abstract concepts arrived at by identifying a common quality in a broad class of phenomena. (If the more mathematically learned have more precise terminology, I’m all ears.) Mathematical manipulators are a special abstract case of causation: it is the actions and interactions implicit in forming and transforming aggregates of quantities.

We then are able to shift mental gears-- ie: enter the Platonic orientation of consciousness– and create a fictional realm with these abstract entities and their abstract interactions. Once we have the abstract fundamental particles of our fictional realm (quantities), and we have the principles of interaction (mathematical manipulators), we are ready to set this model universe in motion, observing and identifying the general rules of interaction as we go.

It is here that mathematics becomes deductive. Once we have identified the general rules of this model universe we call mathematics, it becomes possible to deduce particular truths from general principles.

Now, we can apply inductive reasoning to connect mathematical truths to real world phenomena; thus completing the circle. After a long course of inductive and deductive reasoning, we are back to apply our refined tool to the world in which its raw elements were first discovered.

I think you have the Big Picture of math right. Another nicely done post.

Of course, the sets of rules for the manipulation of quantities need not produce results that have anything to do with the real world, so math has no concept of quantity without induction and it has no known consequences without inductive testing for application after the manipulations of the quantities are completed. So, math both starts with induction and ends with induction.

This sure gives induction a central role in math, at least for anyone who wants it to play a role in understanding reality.

Phil Coates -- and whoever else wants to weigh in on this -- in your considerable experience of hearing Rand and Peikoff (and others) lecture about induction, deduction, and mathematics, do you recall if any of them ever conceded a significant role in mathematics for induction?

REB

Only the finished published product in mathematics is essentially deductive. One publishes theorems and proofs and accompanying discussions.

_Concept_formation_ in mathematics is inductive.

How does one arrive at a suspicion that a particular method might work or a particular proposition might be true, before one sets to work trying to prove it? That's not deductive.

Thanks, everyone, for your input. I agree, and have long believed, that induction is far more important in mathematics than deduction.* Not that deduction is insignificant, but that without induction mathematics -- like any other discipline -- simply could not get off the ground.

Leonard Peikoff in various lectures has characterized mathematics as being essentially deductive and thus unlike the physical sciences in its basic method. He obviously does not know a great deal about mathematics, or he would never have made such claims.

Thankfully, people better versed in mathematics are correcting the record at ARI. Pat Corvini presented a course (probably in 2005) called "The Crisis of Principles in Greek Mathematics," and the blurb in the Ayn Rand Book Store Catalog says that Dr. Corvini "demonstrates that the actual history supports a proper view of mathematics as an inductive science."

So, quod erat demonstrandum for that little controversy!

REB

*There are a number of essays on my website supporting this view, but the best one to read (for relating to me as a person/intellectual) is "Confessions of a Would-Be Mathematician" at this location:

Unposted as yet is another brief piece I wrote in 1989 called "George Polya and Creative Mathematics." I'll share it separately in a post directly below this one.

My interest in mathematics (my original college major, before I dumped it for music) is being rekindled these days. I’ve become infatuated with George Polya, who is (was?) a professor at Stanford. I don’t have his famous book How to Solve It, but I do have both volumes of Mathematics and Plausible Reasoning—A Guide to the Art of Plausible Reasoning, 1954 (which contain a lot on induction and analogy) and both volumes of Mathematical Discovery—On Understanding, Leaning, and Teaching Problem Solving, 1965 (which focus on doing problems and then thinking about the means and methods you use to do them).

Because it is so closely related to my Pythagorean paper and the tinkering I did in high school on binomial expansions, I am fascinated with Polya’s discussions of “methods of guessing at mathematical truths and solutions.” It feels like he’s talking directly about me when he says a truly creative mathematician is a good guesser first and a good prover afterward. That’s certainly the pattern of my process of discovery. Very much the way my mind works. My “psycho-epistemology.” After I finish digesting all he has to say—and finish my Pythagorean project—I intend to do an essay comparing “rational” mathematics, using deduction-demonstrative reasoning-proof, with “empirical” mathematics, using induction-plausible reasoning-guessing. Both can be ways of discovering new knowledge, but (I believe) the latter is much more likely to be productive. Also, although I haven’t yet decided how, I will relate these to Koestler’s “bisociation” as well as so-called “serendipity” or chance (as in: “chance favors the prepared mind”).

In a chapter on mathematical research entitled “Guessing and Scientific Method,” Polya says that the examples and remarks therein “reveal an aspect of mathematics which is as important as it is rarely mentioned: mathematics appears here as a close relative to the natural sciences, as a sort of ‘observational science’ in which observation and analogy may lead to discoveries.”

That’s what I like best about mathematics: the opportunity to be a scientist, to explore and discover. In general, as a matter of act, I am powerfully drawn to investigating, to finding the hidden, deep truth—whether on the concrete level (tracing “lost” people in the past or present, understanding what makes a person “tick”) or the abstract level (philosophy, psychology, mathematics, etc.). It’s not just curiosity, wanting knowledge and understanding. It’s more like being a hunter, stalking the elusive, hidden fact or essence. My various strengths, such as logic or ingenuity or abstract ability, all seem channeled into my desire to track down something and say, “Gotcha!”

As a materials scientist and materials problem-solver, I have often told people that a great part of my work is that of a materials detective. Henceforth, I think I will say that I am both a detective and a hunter. I understand what you are saying here very well. I love your enthusiasm and thirst for knowledge and just plain figuring things out.

One of the great problems I have with the often stated claim that science is the use of the scientific method and that method is the formation of a testable hypothesis by checking to see if its predictions are true, is that this fails to address the realm of scientific knowledge and how one is to come up with hypotheses worthy of expending one's effort and time on checking them out. It takes a lot of careful observation and the organization of a great many facts, such as the nature of entities and their relationships with one another, before one can choose the hypotheses to test with any efficiency. Once one sets out to test the hypothesis, how does one set up a good experiment? Again, one has to know a lot about reality to control the experimental conditions and to have any idea about errors.

Science requires constant observation, organization of those observations, the development of reasonable analogies between similar kinds of entities and events, the use of the scientific method, and the management of limited resources. All of these activities are managed constantly or at least very frequently and they must be used iteratively in support of one another. The scientist is a hunter, a detective, an explorer, a mathematician, a manager, a team member, an advisor, and an improviser. And he is kid in the most incredible play pen full of ingenious toys to study essentially anything he sets his mind to. Sub-atomic particles, stars and galaxies, atoms that organize themselves in many different ways and degrees, molecules that grow and maybe even replicate themselves, living organisms, clouds with incredible patterns, ornery viruses, cells expert in fluid management, incredible optic nerves, much more incredible brains. This is indeed a benevolent universe. It allows us several decades worth of playing the most fascinating games seeking to understand parts of its beautiful complexity.

The scientist is a hunter, a detective, an explorer, a mathematician, a manager, a team member, an advisor, and an improviser. And he is kid in the most incredible play pen full of ingenious toys to study essentially anything he sets his mind to.

You forgot philosopher. Or is that what you meant by "kid in the most incredible play pen full of ingenious toys to study essentially anything he sets his mind to?"

Sure Paul, it would be a good idea to add philosopher, since one's philosophy is always wrapped up in doing science. In fact, it is interesting to observe many scientists with very irrational explicit philosophies utilize very different and more rational philosophies in their actual work.

As an undergraduate, I took a course called The Philosophy of Science. It was almost entirely about logical positivist philosophy. This struck me as having very little to do with how scientists actually did scientific thinking and problem-solving. Logical positivism was totally inadequate for this purpose. I suppose now that scientists are simply viewed as a certain group of analytical thinkers who perform group thought and science emerges, but it is really not a description of the reality of any other group. If every group has their own politics and their own emotions then is it not necessary that every group have their own science?

A thought just struck me. If you are a Christian scientist, then you believe in a god who causes miracles. It must be tempting for the Christian scientist who has failed to find an explanation for a phenomena in physics to simply proclaim the phenomena a miracle of god. But they do not do this. Does this mean that they do not really believe in a god of miracles or does this mean they think god is tired and worn out and cannot make miracles any more? Of course, I think the answer is that god knows he cannot put miracles over on scientists the way he could more primitive man. It probably also does not help that reporters are now quickly on the scene and the real happening is likely to be written down. This makes it harder for churchmen to embellish the event later. It really was very convenient for the sake of miracles after the Pope had forbid the common people to read the Bible, with the result that soon few common folk could read or write. Only churchmen did and they had a certain bias. Miracles could then flourish.

I would observe that mathematics operates as systems of essentially logical rules applied to abstract quantitities, wherein there is a realm in which mathematics is deductive. The applicability of any given set of mathematical rules to informing us about reality is entirely inductive, however. In other words, we only know whether a theory of mathematics rules can be applied to order and explain our knowledge of real entities and their actions on an inductive basis. Without informing our use of mathematics inductively, math is just a game people play for their entertainment. It becomes useful and essential only when we test its use inductively.

So, if you define mathematics very narrowly, you can say it is deductive and not inductive. If you wish to include its use for describing reality in a generous measure, then it is heavily inductive as well as deductive. Academic mathematics departments can be very oriented toward only deductive concerns, but an Applied Mathematics department, or engineers, or most scientists, must be very concerned about the inductive evaluative side of mathematics. Since I am not aware that Peikoff has told us where this boundary is with respect to his statements, I would not quibble with him on this issue. However, I am biased in the direction of thinking that math has its great value as a tool for organizing, predicting, and explaining knowledge about reality. There is a caveat here with respect to explaining reality however, since it really may only explain in the context of organizing and predicting in the context that it works well with respect to our inductive knowledge. Of course, others will say that applying math is the realm of science, but there is plenty of room for applied mathematicians and for mathematical scientists in some sort of boundary area.

If I remember correctly, in Introduction to Objectivist Epistemology, Ayn Rand defined

mathematics as the "science of measurement". At the time I read that definition many years

ago, it struck me as very interesting and profound in its simplicity, but something about it didn't

"feel" completely right to me.

Now, based on the above, I think I realize what it was that didn't feel right. It would seem to me

that "science of measurement" might be a good definition for "applied mathematics". i.e., the

inductive use of mathematics to describe reality.

But the part of mathematics which is "oriented toward only deductive concerns", e.g., proving

theorems, exploring mathematical areas that noone has (yet) tried to apply to the "real world",

is there any way that that can be regarded as the "science of measurement"? If so, what is it

that is being measured? (Nothwithstanding the fact that sometimes mathematics that is

developed in a purely theoretical context is later found to be applicable to some aspect of reality).

I am wondering whether Rand intended her definition to refer to all of the activity that goes

under the heading of "mathematics", or just to applied mathematics.

I think the confusion arises while induction and deduction refer to different things in mathematics. In essence mathematics is completely deductive, but that doesn't mean that human beings don't use inductive reasoning to find the results. But afterwards the inductive scaffolding is removed and the deductive building can stand on its own. Otherwise we wouldn't have a proof, only a conjecture.

Charles:

So, if you define mathematics very narrowly, you can say it is deductive and not inductive. If you wish to include its use for describing reality in a generous measure, then it is heavily inductive as well as deductive.

But in that case it isn't the mathematics that is inductive, but the physics. For example, if I want to know what the geometry of space is, I'll have to do experiments to test whether Euclidean geometry gives the best description or another geometry. But a geometry itself like Euclid's system is a completely deductive system, although Euclid may have arrived at his system inductively by observing things in the real world that seem to be well described by his abstract system. There is always an interaction between physics and mathematics, where physical observations may be the inspiration for mathematical theories which get a life of their own, and which (sometimes much later) may give useful new results to be used in physical theories. But mathematics, whether pure or applied is a deductive system; it is the question what kind of mathematical model or theory fits our physical world best that asks for inductive reasoning.

I am doing an in-depth reading of The Contested Legacy of Ayn Rand by David Kelley and I came across an extremely interesting quote about tying deductive and inductive reasoning to fundamental axioms (pp. 81-82):

The law of identity, which says that a thing must have a specific and non-contradictory nature, is the basis for all deductive reasoning. The law of causality, which says it a thing must act in accordance with its nature, is the basis of all inductive reasoning.

Using Rand's idea of concept formation, mathematics can be seen as deriving from the concept of a single unit (the number 1), where its distinguishing characteristic is "existent." All existents - perceived or imagined - are included in the concept, but eliminated from the concept's distinguishing characteristic. "Something (or anything) that exists" so to speak is the sensory referent for the concept "1". This connects the mind's conceptual integrating capacity to sensory evidence (since there are oodles of existents we perceive all life long). The single unit is arrived at through induction and integration. Then "1" can stand for any particular existent, or even no existent at all except itself (since the concept "1" also is an existent), and still retain the law of identity (the fundamental axiom). Once identity is in place without all those messy existential referents, you have an all-inclusive concept with a non-contradictory nature and the rest is deduction. To paraphrase:

1 is 1

I also see this concept as easy to teach at a very early age without the technical language, which is why it is so universal among humans.

Put in two other derivative ideas from this all-inclusive concept - a unit for no existent at all (0) and a unit for more than one existent (2) - and you start to develop a mathematical system. Then make a unit for something that "eats" a unit (negative number), a bunch of units for more than 2, and devise some ways to manipulate them. Then the sky's the limit.

All this is deduction. (If a concept, retaining the law of identity, can represent something that exists, then by its nature as derived from induction and integration, such a concept, retaining the law of identity, also can be projected to represent nothing at all - "0". This deductive reasoning can be applied to deriving all the other units.)

Interestingly, I see the basic actions of addition, subtraction, division and multiplication as deriving initially from induction - from physically doing these actions to things (or observing something/someone else doing them).

I see no problem at all in using a mathematical system in inductive reasoning from there. I don't see a way to construct mathematics without reference to a single unit, though, and that is arrived at through induction plus integration like all basic concepts are.

There are many good and informed posts on this thread. I thought some of you might like to see the work that was done on induction, deduction, and their places in mathematics and science in my journal Objectivity (1990-98). I will show you the Abstracts of the pertinent essays. If anyone would like to have a copy of any one of these essays, just let me know.

ABSTRACTS

“Philosophy of Mathematics” by Merlin Jetton

Volume 1, Number 2, Pages 1–32

First is Plato’s account of the deductive discipline of Greek mathematics, the ontology of its objects, and the necessity of its truths. Then is Aristotle’s account of mathematical objects as essences abstracted from empirical objects, his treatment of mathematical necessity as hypothetical, and his influence on Euclid’s organization of geometry.

Jetton next turns to Leibniz and Hume, then Kant. Here is Leibniz’s treatment of mathematical truths as logical truths. Here is Hume’s account of mathematical relations as relations of ideas, made necessarily true only by acts of mind, not by their occasions in nature.

Jetton sets out in some detail Kant’s cast of pure mathematics as at once synthetic and a priori. He recounts Kant’s marriage of construction in pure Euclidean geometry to the mind’s automatic synthesis of manifold appearance into perceptual experience. The discovery of Non-Euclidean geometries in the century after Kant poses problems for Kant’s view of the discipline of geometry and its bearing on the physical world in experience. Jetton addresses these problems.

Next under the spotlight is the empiricist theory of mathematical knowledge authored by Mill. Then there appears the logicism of Frege and Russell, the intuitionism of Brouwer and Heyting, and the formalism of Hilbert. The implications of the limitative logical theorems discovered by Gödel are discussed.

Jetton concludes with his own resulting view of mathematical knowledge. His view is strongly empiricist—stressing the epistemological precedence of applied mathematics over pure mathematics—but modulated by aspects of conceptual consciousness underrated in traditional empiricism.

“Mathematic Empiric” by Daniel Ust

Volume 1, Number 6, Pages 55–71

Philip Kitcher’s contemporary empiricist theory of mathematical knowledge is here surveyed. Kitcher argues that mathematical knowledge need not be independent of experience, that is, it need not be a priori, in order to be true in all possible contexts. Knowledge is true belief gotten by a warranted process. No warranting process, such as “intuition,” can be available independently of experience nor produce true and warranted belief independently of experience.

Kitcher argues against the three basic types of a priori-ism in mathematical philosophy: realism, constructivism, and conceptualism. He has a positive proposal for how the entirety of modern mathematics, pure as well as applied, has been developed from empirical origins.

"Mathematics and Intuition" by Kathleen Touchstone

Volume 2, Number 4, Pages93-182

Touchstone explores elements of intuition in the invention and use of mathematics. Within the sweep of this exploration are: Penelope Maddy's contemporary realism in mathematics, which attempts to bring sets into the physical world and to tie mathematical intuitions to perception; Ray Jackendoff's case for a universal innate grammar and the intuitive nature of thought; Reber's work on implicit learning and tacit knowledge; Gelman and Gallistel's research on the child's concept of number; computational perspectives on mental representations; Stephen Grossberg's neural network modeling of perception and learning

“Induction on Identity” by Stephen Boydstun

Part 1Volume 1, Number 2, Pages 33–46

The Aristotelian and Leibnizian roles of non-contradiction and identity in metaphysics and in deductive logic are reviewed. Beyond those roles, Boydstun proposes that Rand’s existential law of identity can fully justify inductive inference.

The two types of induction articulated by Aristotle are rehearsed. These are the abstractive induction and the ampliative induction. Rand had noted that the integration of facts into concepts is a type of induction. This is abstractive induction à la Rand, and Boydstun resolves this type into two components: the bare recursive induction we use in mathematical induction and the ampliative induction needed for the construction of concepts adequate to the concrete existents in the world.

Part 2 Volume 1, Number 3, Pages 1–56

Boydstun examines the critiques of the rationality of induction put forth in the fourteenth century by Nicolaus of Autrecourt and in the eighteenth century by David Hume. The stage for Nicolaus was set by Ockham. On this stage were the metaphysical and logical platforms for arguing that there can be no logically necessary connection between distinct existents. Nicolaus’ hour on the stage articulates how to hold fast to identity, non-contradiction, and the existential presentations of immediate perception, while barring any logically justifiable inference to the existence of material substance. Boydstun disputes Nicolaus’ account of our experience of substance and of our rational inferences to the existence and character of substance not directly experienced. The defense of our knowledge of substance here marshals logical considerations, findings of modern developmental psychology, and the history of modern science.

Hume’s accounts of our experience of cause and effect and of our reasonings to cause or to effect are closely examined and roundly criticized. Various statements of the law of causality in the history of philosophy are recounted and assessed, with due consideration of modern physics. A version of the law of causality logically supportable by Rand’s rich principle of identity is formulated. It does not require that in a given circumstance a given kind of thing could do only the same single thing on repeated trials. It is argued also that Rand’s principle of identity is the broad base of Mill’s methods of induction and of the hypothetico-deductive method of science.

Beyond the corrected law of causality, Boydstun formulates a “principle of substantive propagation.” This is an application of Rand’s principle of identity to all alteration and constancy in time. He argues that the principle of substantive propagation is a fundamental justified justification for our inductive causal inferences and indeed all of our modes of scientific explanation. He concludes with a proposal of how predication can be cast as a triple-identity abstract form, a derivative of Rand’s fundamental thesis that existence is identity.

Here is "About Objectivity" which will be a feature of the Objectivity_Archive site that is being constructed. This will tell you some of the story behind the journal, and it includes a complete list of compositions that were published in the journal.

About Objectivity

Objectivity is a journal of metaphysics, epistemology, and theory of value informed by modern science. It was the creation of Stephen Boydstun. It was a hardcopy journal, for subscribers, published from 1990 to 1998. Its authors were both professional academics and independent scholars.

. . . .

During the years of making Objectivity, Stephen was slowly dying of AIDS. He created the journal in the desolation of the death of his beloved Jerry. The making of the journal became his lifeline. In 1996 he met Walter Klingler. They are both sustained against AIDS by advances in medical science, and they continue “two hearts beating each to each.”

Sometimes a kind of glory lights up the mind of a man. ―Steinbeck

Contributors

Charles Wieder ~ Christopher Weed ~ Philip Wagner ~ Paul Vanderveen ~ Daniel Ust Kathleen Touchstone ~ Robert Sweeney ~ Fred Seddon ~ Ray Shelton ~ Peter Saint-André David Ross ~ David Potts ~ Svein Olav Nyberg ~ Eyal Mozes ~ Joseph Mixie ~ Ronald E. Merrill ~ Gary McGath ~ Tibor Machan ~ George Lyons ~ Irfan Khawaja ~ Merlin Jetton Michael Huemer ~ Stephen Hicks ~ James Henderson ~ Ron Harris ~ Thomas Gramstad Jay Friedenberg ~ John Enright ~ Marsha Familaro Enright ~ Paul Enright ~ Rafael Eilon Peter Chriss ~ Philip Coates ~ Stephen Boydstun ~ Roger E. Bissell

To obtain these articles, visit www.bomis.com/objectivity/ and click on the Prices sector. The prices of any and all issues of Objectivity are listed there. Instructions for making payment are also given in that sector.

When you put your check or money order into the mail, send an e-mail to boydstun@rcn.com. Let us know which issues you have ordered and your US Postal Service address. All issues are mailed Priority.

To obtain these articles, visit www.bomis.com/objectivity/ and click on the Prices sector. The prices of any and all issues of Objectivity are listed there. Instructions for making payment are also given in that sector.

When you put your check or money order into the mail, send an e-mail to boydstun@rcn.com. Let us know which issues you have ordered and your US Postal Service address. All issues are mailed Priority.

Thank you for your interest.

Stephen

Hey, you've got to enter the 21st century and accept payment via Paypal!

No matter, I'll probably be ordering a bunch if not all of these...many of the titles look very

~ I believe I have some of your beginning issues (2 yrs worth) of Objectivity somewhere around (I used to be quite organized, but, as of late...). Been meaning to continue the subscription; definitely all were worth reading.

~ The question of this thread is probably best answerable from a deeper area: the relationship ('logically') between Induction and Deduction themselves.

~ I've argued elsewhere that accepting Deduction as a useable 'method' (re ANYthing, much less only 'math') requires that one accept Induction as the base for accepting Deduction's continued useability. Some have argued (unclear as to whether they 'deductively' did so, or 'inductively' did so; they never specified, interestingly) against my view, but, I've not really seen what they regard as a 'flaw' in it. --- Given that, as pointed out by some re 'enumerating' items/entities, the relationships Deducible within math are resultant from Inductions from original observations.

~ Interesting thread.

LLAP

J:D

P.S: I do find it unfortunate that LP has had little to extemporize on these subjects' relationships, but for occasional 'aside' commenting. We're talking the very BASE of 'Epistemology', for Pete's sakes!

"accepting Deduction as a useable 'method' (re ANYthing, much less only 'math') requires that one accept Induction as the base for accepting Deduction's continued useability"

reminds me of something I wrote in the 1991 Objectivity essay "Induction on Identity." I have a suggestion for how to test and develope this idea further, but first I'll display the related stretch of my essay.

You will recall that in that essay I introduced as an existential principle that I called 'the principle of substantive propagation' and that I took the law of identity (in Rand's conception of it) and the law of noncontradiction to be existential principles as well as formal principles. Here is the passage (V1N3 40):

The modern Humean would continue shamelessly: "Yes, Einstein's field equations, Hamilton's principle, and the conservation of energy may have all held up until now, but what about tomorrow? We concede that since the beginning of time there has been nothing cosmically special about present times per se, but what about present times per se in the future? And again, what about this principle of substantive propagation, this principle that the way things will be grow out of the way things have been? That may have been true yesterday, but what about tomorrow?"

I shall give a negative argument, then a positive argument. If the ground of induction is the law of identity and no principle is entirely independent of this law, then my arguments should be ultimately circular---well, perhaps spiral, perhaps widening for the positive argument and narrowing for the negative. At any rate of curvature, we may count our circularity on this issue benign, even virtuous. This is not meant to shield my arguments from other types of rational criticism.

The negative argument is this: The principle that the ways things will be grow out of the ways things have been is not a bald analytic statement. It is not a stupid tautology. One could go ahead and say "perhaps it is not always the case that the ways things will be grow out of the ways things have been" without having misunderstood the simple meanings of the terms in the principle. Suppose the principle were false. Would the principle of noncontradiction then be true tomorrow? If the principle of noncontradiction is true tomorrow without having been an identity in time with itself today, then it is a radical, Humean contingent truth. Then, in the Humean mentality, it could be false day after tomorrow. Suppose the principle of noncontradiction be false tomorrow. Then was it true today? Suppose the principle of noncontradiction is true for any tomorrow but only contingently so. Better yet, suppose we just return to reality. There we find all the necessity worth having. There we find the principle of noncontradiction and our principle of substantive propagation always true.

[Then comes my positive argument, 41-43]

The suggestion I wanted to mention is this. Look into mathematical induction and its relations to logical principles in modern deductive logic. Mathematical induction is essential to all ordered numbers, and these sequences have application not only to unfoldings in time but elsewhere. Mathematics presupposes deductive logic, but then adds its own further postulates. Is mathematical induction (the weak and strong forms of MI and the least number principle follow from one another) one of those additional postulates of mathematics? Or is it part of deductive logic, having uses in logic itself? If it is has useful work to do in deductive logic itself, what specifically?

I think the text by J. L. Bell and M. Machover A Course in Mathematical Logic may have the answers to those questions.

Your comments are correct if we limit our interest to the body of knowledge only, but as you point out, the practice of mathematics often involves an integration of math and physics. When the mathematian sets out to develop a new mathematical area for research, he usually wishes to develop a mathematics that has the applicability to the real world that is required for it to have the ability to prove anything. He has to build that in, or at least try to. When that area of mathematical investigation results in some proofs about some modeled aspect of the real world, one still wants to check observations and see if the proof is consistent with the observation. Yes, you can theoretically say the mathematician does only the deductive calculations and reasonings, but if he really does only that, then he may as well be playing chess or some other game. Generally, he draws a salary and has some responsibility to do something useful. Often, he teaches, but why should anyone study math if it has no applications? So, in general mathematians try to develop rules and results that do have some applicability to explaining the world about us. Of course, one could develop rules and apply them mathematically with no interest and no regard for reality and it could be said that you were doing mathematics. But, it would be rather like saying someone was a scientist because they were constantly proposing hypotheses and testing them against reality even though every test showed every hypothesis was false and one was left scratching one's head wondering why anyone would ever even have thought that these hypotheses were worth testing.

So the question is this: Does mathematics contain within its field concern for its usefulness or do we assume that the field should have no responsibility and no interest in its usefulness? Should a university have mathematicians who only play arbitrary deductive games and a physics department which points out when their results chance to be useful and when they are useless? I am willing to allow mathematicians to be concerned with the usefulness of their field of study and even develop proofs of physics in the process. I would also allow the physicists and engineers to use their results and even contribute to the field of mathematics as they wish.

I think the confusion arises while induction and deduction refer to different things in mathematics. In essence mathematics is completely deductive, but that doesn't mean that human beings don't use inductive reasoning to find the results. But afterwards the inductive scaffolding is removed and the deductive building can stand on its own. Otherwise we wouldn't have a proof, only a conjecture.

Charles:

So, if you define mathematics very narrowly, you can say it is deductive and not inductive. If you wish to include its use for describing reality in a generous measure, then it is heavily inductive as well as deductive.

But in that case it isn't the mathematics that is inductive, but the physics. For example, if I want to know what the geometry of space is, I'll have to do experiments to test whether Euclidean geometry gives the best description or another geometry. But a geometry itself like Euclid's system is a completely deductive system, although Euclid may have arrived at his system inductively by observing things in the real world that seem to be well described by his abstract system. There is always an interaction between physics and mathematics, where physical observations may be the inspiration for mathematical theories which get a life of their own, and which (sometimes much later) may give useful new results to be used in physical theories. But mathematics, whether pure or applied is a deductive system; it is the question what kind of mathematical model or theory fits our physical world best that asks for inductive reasoning.

When the mathematian sets out to develop a new mathematical area for research, he usually wishes to develop a mathematics that has the applicability to the real world that is required for it to have the ability to prove anything.

That is greatly exaggerated. SOME mathematicians may usually have in mind

applications that non-mathematicians can use, but often applications are to others

areas of mathematics, which in turn are applied to other areas of mathematics,

etc., and applications to the "real" world may show up 50 years later.

Yes, you can theoretically say the mathematician does only the deductive calculations and reasonings,

Not true. One CHECKS the correctness of the end result by using only deduction.

How one GETS the end result is another matter. Concept formation, in particular,

is involved, and is an inductive process. And it's an vitually overused cliche to say that

mathematicians rely heavily on "intuition" (although there might not be much agreement

on how to define that term, if anyone were to examine it).

So, in general mathematians try to develop rules and results that do have some applicability to explaining the world about us.

Only if you construe that in the light of my qualifications above. -- Mike Hardy

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## Philip Coates 1

> Peikoff therein talked about mathematics being atypical (epistemologically) because it was deductive...Has he modified this view since 1983?

Roger, this is the kind of thing I would have qualified in my mind with the word 'largely' or 'basically', had i heard it, assuming he realized many concepts were abstracted from (induced) from concrete instances and he was 'essentializing'.

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## SaulOhio 0

I would definitely say that numbers are taught inductively. You show a child five blocks, tell them this is "five". You show the child five sticks, say that is "five". You repeat this with different numbers and objects till the child gets the idea of quantity.

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## Charles R. Anderson 0

Roger,

I cannot address your question with respect to Peikoff and mathematics as a deductive field directly.

I would observe that mathematics operates as systems of essentially logical rules applied to abstract quantitities, wherein there is a realm in which mathematics is deductive. The applicability of any given set of mathematical rules to informing us about reality is entirely inductive, however. In other words, we only know whether a theory of mathematics rules can be applied to order and explain our knowledge of real entities and their actions on an inductive basis. Without informing our use of mathematics inductively, math is just a game people play for their entertainment. It becomes useful and essential only when we test its use inductively.

So, if you define mathematics very narrowly, you can say it is deductive and not inductive. If you wish to include its use for describing reality in a generous measure, then it is heavily inductive as well as deductive. Academic mathematics departments can be very oriented toward only deductive concerns, but an Applied Mathematics department, or engineers, or most scientists, must be very concerned about the inductive evaluative side of mathematics. Since I am not aware that Peikoff has told us where this boundary is with respect to his statements, I would not quibble with him on this issue. However, I am biased in the direction of thinking that math has its great value as a tool for organizing, predicting, and explaining knowledge about reality. There is a caveat here with respect to explaining reality however, since it really may only explain in the context of organizing and predicting in the context that it works well with respect to our inductive knowledge. Of course, others will say that applying math is the realm of science, but there is plenty of room for applied mathematicians and for mathematical scientists in some sort of boundary area.

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## Paul Mawdsley 2

Could it not be said that the concept of "quantity" itself is arrived at inductively? We arrive at the abstract concept of "quantity" by identifying a common quality in a broad class of entities– ie: all entities-- and subtracting all other qualities from observed entities. It is no different to arriving at the concept of causality by identifying the common quality in a broad class of phenomena– actions and interactions of entities-- and subtracting all other observed qualities.

Also I tend to see the basic mathematical manipulators – eg: adding, subtracting, etc.– as abstract concepts arrived at by identifying a common quality in a broad class of phenomena. (If the more mathematically learned have more precise terminology, I’m all ears.) Mathematical manipulators are a special abstract case of causation: it is the actions and interactions implicit in forming and transforming aggregates of quantities.

We then are able to shift mental gears-- ie: enter the Platonic orientation of consciousness– and create a fictional realm with these abstract entities and their abstract interactions. Once we have the abstract fundamental particles of our fictional realm (quantities), and we have the principles of interaction (mathematical manipulators), we are ready to set this model universe in motion, observing and identifying the general rules of interaction as we go.

It is here that mathematics becomes deductive. Once we have identified the general rules of this model universe we call mathematics, it becomes possible to deduce particular truths from general principles.

Now, we can apply inductive reasoning to connect mathematical truths to real world phenomena; thus completing the circle. After a long course of inductive and deductive reasoning, we are back to apply our refined tool to the world in which its raw elements were first discovered.

Paul

(I hope you don't mind a layman's point of view.)

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## Charles R. Anderson 0

Paul,

I think you have the Big Picture of math right. Another nicely done post.

Of course, the sets of rules for the manipulation of quantities need not produce results that have anything to do with the real world, so math has no concept of quantity without induction and it has no known consequences without inductive testing for application after the manipulations of the quantities are completed. So, math both starts with induction and ends with induction.

This sure gives induction a central role in math, at least for anyone who wants it to play a role in understanding reality.

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## Mike Hardy 0

Only the finished published product in mathematics is essentially deductive. One publishes theorems and proofs and accompanying discussions.

_Concept_formation_ in mathematics is inductive.

How does one arrive at a suspicion that a particular method might work or a particular proposition might be true, before one sets to work trying to prove it? That's not deductive.

-- Mike Hardy

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## Roger Bissell 15

AuthorThanks, everyone, for your input. I agree, and have long believed, that induction is far more important in mathematics than deduction.* Not that deduction is insignificant, but that without induction mathematics -- like any other discipline -- simply could not get off the ground.

Leonard Peikoff in various lectures has characterized mathematics as being essentially deductive and thus unlike the physical sciences in its basic method. He obviously does not know a great deal about mathematics, or he would never have made such claims.

Thankfully, people better versed in mathematics are correcting the record at ARI. Pat Corvini presented a course (probably in 2005) called "The Crisis of Principles in Greek Mathematics," and the blurb in the Ayn Rand Book Store Catalog says that Dr. Corvini "demonstrates that the actual history supports a proper view of mathematics as an inductive science."

So,

quod erat demonstrandumfor that little controversy!REB

*There are a number of essays on my website supporting this view, but the best one to read (for relating to me as a person/intellectual) is "Confessions of a Would-Be Mathematician" at this location:

http://members.aol.com/REBissell/mmmConfes...hematician.html

Unposted as yet is another brief piece I wrote in 1989 called "George Polya and Creative Mathematics." I'll share it separately in a post directly below this one.

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## Roger Bissell 15

AuthorGeorge Polya and Creative MathematicsBy Roger E. BissellDecember 31, 1989

My interest in mathematics (my original college major, before I dumped it for music) is being rekindled these days. I’ve become infatuated with George Polya, who is (was?) a professor at Stanford. I don’t have his famous book

How to Solve It, but I do have both volumes ofMathematics and Plausible Reasoning—A Guide to the Art of Plausible Reasoning, 1954 (which contain a lot on induction and analogy) and both volumes ofMathematical Discovery—On Understanding, Leaning, and Teaching Problem Solving, 1965 (which focus on doing problems and then thinking about the means and methods you use to do them).Because it is so closely related to my Pythagorean paper and the tinkering I did in high school on binomial expansions, I am fascinated with Polya’s discussions of “methods of guessing at mathematical truths and solutions.” It feels like he’s talking directly about me when he says a truly creative mathematician is a good guesser first and a good prover afterward. That’s certainly the

patternof my process of discovery. Very much the way my mind works. My “psycho-epistemology.” After I finish digesting all he has to say—and finish my Pythagorean project—I intend to do an essay comparing “rational” mathematics, using deduction-demonstrative reasoning-proof, with “empirical” mathematics, using induction-plausible reasoning-guessing. Bothcanbe ways of discovering new knowledge, but (I believe) the latter is much morelikelyto be productive. Also, although I haven’t yet decided how, I will relate these to Koestler’s “bisociation” as well as so-called “serendipity” or chance (as in: “chance favors the prepared mind”).In a chapter on mathematical research entitled “Guessing and Scientific Method,” Polya says that the examples and remarks therein “reveal an aspect of mathematics which is as important as it is rarely mentioned: mathematics appears here as a close relative to the natural sciences, as a sort of ‘observational science’ in which observation and analogy may lead to discoveries.”

That’s what I like

best about mathematics: the opportunity to be ascientist, to explore and discover. In general, as a matter of act, I am powerfully drawn to investigating, to finding the hidden, deep truth—whether on the concrete level (tracing “lost” people in the past or present, understanding what makes a person “tick”) or the abstract level (philosophy, psychology, mathematics, etc.). It’s not just curiosity, wanting knowledge and understanding. It’s more like being ahunter, stalking the elusive, hidden fact or essence. My various strengths, such as logic or ingenuity or abstract ability, all seem channeled into my desire to track down something and say, “Gotcha!”## Link to post

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## Charles R. Anderson 0

As a materials scientist and materials problem-solver, I have often told people that a great part of my work is that of a materials detective. Henceforth, I think I will say that I am both a detective and a hunter. I understand what you are saying here very well. I love your enthusiasm and thirst for knowledge and just plain figuring things out.

One of the great problems I have with the often stated claim that science is the use of the scientific method and that method is the formation of a testable hypothesis by checking to see if its predictions are true, is that this fails to address the realm of scientific knowledge and how one is to come up with hypotheses worthy of expending one's effort and time on checking them out. It takes a lot of careful observation and the organization of a great many facts, such as the nature of entities and their relationships with one another, before one can choose the hypotheses to test with any efficiency. Once one sets out to test the hypothesis, how does one set up a good experiment? Again, one has to know a lot about reality to control the experimental conditions and to have any idea about errors.

Science requires constant observation, organization of those observations, the development of reasonable analogies between similar kinds of entities and events, the use of the scientific method, and the management of limited resources. All of these activities are managed constantly or at least very frequently and they must be used iteratively in support of one another. The scientist is a hunter, a detective, an explorer, a mathematician, a manager, a team member, an advisor, and an improviser. And he is kid in the most incredible play pen full of ingenious toys to study essentially anything he sets his mind to. Sub-atomic particles, stars and galaxies, atoms that organize themselves in many different ways and degrees, molecules that grow and maybe even replicate themselves, living organisms, clouds with incredible patterns, ornery viruses, cells expert in fluid management, incredible optic nerves, much more incredible brains. This is indeed a benevolent universe. It allows us several decades worth of playing the most fascinating games seeking to understand parts of its beautiful complexity.

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## Paul Mawdsley 2

Charles, you wrote:

You forgot philosopher. Or is that what you meant by "kid in the most incredible play pen full of ingenious toys to study essentially anything he sets his mind to?"

Paul

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## Charles R. Anderson 0

Sure Paul, it would be a good idea to add philosopher, since one's philosophy is always wrapped up in doing science. In fact, it is interesting to observe many scientists with very irrational explicit philosophies utilize very different and more rational philosophies in their actual work.

As an undergraduate, I took a course called The Philosophy of Science. It was almost entirely about logical positivist philosophy. This struck me as having very little to do with how scientists actually did scientific thinking and problem-solving. Logical positivism was totally inadequate for this purpose. I suppose now that scientists are simply viewed as a certain group of analytical thinkers who perform group thought and science emerges, but it is really not a description of the reality of any other group. If every group has their own politics and their own emotions then is it not necessary that every group have their own science?

A thought just struck me. If you are a Christian scientist, then you believe in a god who causes miracles. It must be tempting for the Christian scientist who has failed to find an explanation for a phenomena in physics to simply proclaim the phenomena a miracle of god. But they do not do this. Does this mean that they do not really believe in a god of miracles or does this mean they think god is tired and worn out and cannot make miracles any more? Of course, I think the answer is that god knows he cannot put miracles over on scientists the way he could more primitive man. It probably also does not help that reporters are now quickly on the scene and the real happening is likely to be written down. This makes it harder for churchmen to embellish the event later. It really was very convenient for the sake of miracles after the Pope had forbid the common people to read the Bible, with the result that soon few common folk could read or write. Only churchmen did and they had a certain bias. Miracles could then flourish.

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## MBM 0

If I remember correctly, in Introduction to Objectivist Epistemology, Ayn Rand defined

mathematics as the "science of measurement". At the time I read that definition many years

ago, it struck me as very interesting and profound in its simplicity, but something about it didn't

"feel" completely right to me.

Now, based on the above, I think I realize what it was that didn't feel right. It would seem to me

that "science of measurement" might be a good definition for "applied mathematics". i.e., the

inductive use of mathematics to describe reality.

But the part of mathematics which is "oriented toward only deductive concerns", e.g., proving

theorems, exploring mathematical areas that noone has (yet) tried to apply to the "real world",

is there any way that that can be regarded as the "science of measurement"? If so, what is it

that is being measured? (Nothwithstanding the fact that sometimes mathematics that is

developed in a purely theoretical context is later found to be applicable to some aspect of reality).

I am wondering whether Rand intended her definition to refer to all of the activity that goes

under the heading of "mathematics", or just to applied mathematics.

I'd be interested in any thoughts on the above.

MBM

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## Dragonfly 6

I think the confusion arises while induction and deduction refer to different things in mathematics. In essence mathematics is completely deductive, but that doesn't mean that human beings don't use inductive reasoning to find the results. But afterwards the inductive scaffolding is removed and the deductive building can stand on its own. Otherwise we wouldn't have a proof, only a conjecture.

Charles:

But in that case it isn't the mathematics that is inductive, but the physics. For example, if I want to know what the geometry of space is, I'll have to do experiments to test whether Euclidean geometry gives the best description or another geometry. But a geometry itself like Euclid's system is a completely deductive system, although Euclid may have arrived at his system inductively by observing things in the real world that seem to be well described by his abstract system. There is always an interaction between physics and mathematics, where physical observations may be the inspiration for mathematical theories which get a life of their own, and which (sometimes much later) may give useful new results to be used in physical theories. But mathematics, whether pure or applied

isa deductive system; it is the question what kind of mathematical model or theory fits our physical world best that asks for inductive reasoning.## Link to post

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## Michael Stuart Kelly 1,653

I am doing an in-depth reading of

The Contested Legacy of Ayn Randby David Kelley and I came across an extremely interesting quote about tying deductive and inductive reasoning to fundamental axioms (pp. 81-82):Using Rand's idea of concept formation, mathematics can be seen as deriving from the concept of a single unit (the number 1), where its distinguishing characteristic is "existent." All existents - perceived or imagined - are included in the concept, but eliminated from the concept's distinguishing characteristic. "Something (or anything) that exists" so to speak is the sensory referent for the concept "1". This connects the mind's conceptual integrating capacity to sensory evidence (since there are oodles of existents we perceive all life long). The single unit is arrived at through induction and integration. Then "1" can stand for any particular existent, or even no existent at all except itself (since the concept "1" also is an existent), and still retain the law of identity (the fundamental axiom). Once identity is in place without all those messy existential referents, you have an all-inclusive concept with a non-contradictory nature and the rest is deduction. To paraphrase:

1 is 1

I also see this concept as easy to teach at a very early age without the technical language, which is why it is so universal among humans.

Put in two other derivative ideas from this all-inclusive concept - a unit for no existent at all (0) and a unit for more than one existent (2) - and you start to develop a mathematical system. Then make a unit for something that "eats" a unit (negative number), a bunch of units for more than 2, and devise some ways to manipulate them. Then the sky's the limit.

All this is deduction. (If a concept, retaining the law of identity, can represent something that exists, then by its nature as derived from induction and integration, such a concept, retaining the law of identity, also can be projected to represent nothing at all - "0". This deductive reasoning can be applied to deriving all the other units.)

Interestingly, I see the basic actions of addition, subtraction, division and multiplication as deriving initially from induction - from physically doing these actions to things (or observing something/someone else doing them).

I see no problem at all in using a mathematical system in inductive reasoning from there. I don't see a way to construct mathematics without reference to a single unit, though, and that is arrived at through induction plus integration like all basic concepts are.

Michael

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## Guyau 11

There are many good and informed posts on this thread. I thought some of you might like to see the work that was done on induction, deduction, and their places in mathematics and science in my journal(1990-98). I will show you the Abstracts of the pertinent essays. If anyone would like to have a copy of any one of these essays, just let me know.

ObjectivityABSTRACTS

“Philosophy of Mathematics”by Merlin JettonVolume 1, Number 2, Pages 1–32First is Plato’s account of the deductive discipline of Greek mathematics, the ontology of its objects, and the necessity of its truths. Then is Aristotle’s account of mathematical objects as essences abstracted from empirical objects, his treatment of mathematical necessity as hypothetical, and his influence on Euclid’s organization of geometry.

Jetton next turns to Leibniz and Hume, then Kant. Here is Leibniz’s treatment of mathematical truths as logical truths. Here is Hume’s account of mathematical relations as relations of ideas, made necessarily true only by acts of mind, not by their occasions in nature.

Jetton sets out in some detail Kant’s cast of pure mathematics as at once synthetic and a priori. He recounts Kant’s marriage of construction in pure Euclidean geometry to the mind’s automatic synthesis of manifold appearance into perceptual experience. The discovery of Non-Euclidean geometries in the century after Kant poses problems for Kant’s view of the discipline of geometry and its bearing on the physical world in experience. Jetton addresses these problems.

Next under the spotlight is the empiricist theory of mathematical knowledge authored by Mill. Then there appears the logicism of Frege and Russell, the intuitionism of Brouwer and Heyting, and the formalism of Hilbert. The implications of the limitative logical theorems discovered by Gödel are discussed.

Jetton concludes with his own resulting view of mathematical knowledge. His view is strongly empiricist—stressing the epistemological precedence of applied mathematics over pure mathematics—but modulated by aspects of conceptual consciousness underrated in traditional empiricism.

“Mathematic Empiric”by Daniel UstVolume 1, Number 6, Pages 55–71Philip Kitcher’s contemporary empiricist theory of mathematical knowledge is here surveyed. Kitcher argues that mathematical knowledge need not be independent of experience, that is, it need not be a priori, in order to be true in all possible contexts. Knowledge is true belief gotten by a warranted process. No warranting process, such as “intuition,” can be available independently of experience nor produce true and warranted belief independently of experience.

Kitcher argues against the three basic types of a priori-ism in mathematical philosophy: realism, constructivism, and conceptualism. He has a positive proposal for how the entirety of modern mathematics, pure as well as applied, has been developed from empirical origins.

"Mathematics and Intuition"by Kathleen TouchstoneVolume 2, Number 4, Pages93-182Touchstone explores elements of intuition in the invention and use of mathematics. Within the sweep of this exploration are: Penelope Maddy's contemporary realism in mathematics, which attempts to bring sets into the physical world and to tie mathematical intuitions to perception; Ray Jackendoff's case for a universal innate grammar and the intuitive nature of thought; Reber's work on implicit learning and tacit knowledge; Gelman and Gallistel's research on the child's concept of number; computational perspectives on mental representations; Stephen Grossberg's neural network modeling of perception and learning

“Induction on Identity”by Stephen BoydstunPart 1Volume 1, Number 2, Pages 33–46The Aristotelian and Leibnizian roles of non-contradiction and identity in metaphysics and in deductive logic are reviewed. Beyond those roles, Boydstun proposes that Rand’s existential law of identity can fully justify inductive inference.

The two types of induction articulated by Aristotle are rehearsed. These are the abstractive induction and the ampliative induction. Rand had noted that the integration of facts into concepts is a type of induction. This is abstractive induction à la Rand, and Boydstun resolves this type into two components: the bare recursive induction we use in mathematical induction and the ampliative induction needed for the construction of concepts adequate to the concrete existents in the world.

Part 2Volume 1, Number 3, Pages 1–56Boydstun examines the critiques of the rationality of induction put forth in the fourteenth century by Nicolaus of Autrecourt and in the eighteenth century by David Hume. The stage for Nicolaus was set by Ockham. On this stage were the metaphysical and logical platforms for arguing that there can be no logically necessary connection between distinct existents. Nicolaus’ hour on the stage articulates how to hold fast to identity, non-contradiction, and the existential presentations of immediate perception, while barring any logically justifiable inference to the existence of material substance. Boydstun disputes Nicolaus’ account of our experience of substance and of our rational inferences to the existence and character of substance not directly experienced. The defense of our knowledge of substance here marshals logical considerations, findings of modern developmental psychology, and the history of modern science.

Hume’s accounts of our experience of cause and effect and of our reasonings to cause or to effect are closely examined and roundly criticized. Various statements of the law of causality in the history of philosophy are recounted and assessed, with due consideration of modern physics. A version of the law of causality logically supportable by Rand’s rich principle of identity is formulated. It does not require that in a given circumstance a given kind of thing could do only the same single thing on repeated trials. It is argued also that Rand’s principle of identity is the broad base of Mill’s methods of induction and of the hypothetico-deductive method of science.

Beyond the corrected law of causality, Boydstun formulates a “principle of substantive propagation.” This is an application of Rand’s principle of identity to all alteration and constancy in time. He argues that the principle of substantive propagation is a fundamental justified justification for our inductive causal inferences and indeed all of our modes of scientific explanation. He concludes with a proposal of how predication can be cast as a triple-identity abstract form, a derivative of Rand’s fundamental thesis that existence is identity.

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## Paul Mawdsley 2

Stephen,

Your

Objectivityjournal has some interesting titles. I am very curious.Paul

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## Guyau 11

Here is "About

Objectivity" which will be a feature of theObjectivity_Archivesite that is being constructed. This will tell you some of the story behind the journal, and it includes a complete list of compositions that were published in the journal.AboutObjectivityObjectivityis a journal of metaphysics, epistemology, and theory of value informed by modern science. It was the creation of Stephen Boydstun. It was a hardcopy journal, for subscribers, published from 1990 to 1998. Its authors were both professional academics and independent scholars.. . . .

During the years of making

Objectivity, Stephen was slowly dying of AIDS. He created the journal in the desolation of the death of his beloved Jerry. The making of the journal became his lifeline. In 1996 he met Walter Klingler. They are both sustained against AIDS by advances in medical science, and they continue “two hearts beating each to each.”Sometimes a kind of glory lights up the mind of a man. ―SteinbeckContributorsCharles Wieder ~ Christopher Weed ~ Philip Wagner ~ Paul Vanderveen ~ Daniel Ust Kathleen Touchstone ~ Robert Sweeney ~ Fred Seddon ~ Ray Shelton ~ Peter Saint-André David Ross ~ David Potts ~ Svein Olav Nyberg ~ Eyal Mozes ~ Joseph Mixie ~ Ronald E. Merrill ~ Gary McGath ~ Tibor Machan ~ George Lyons ~ Irfan Khawaja ~ Merlin Jetton Michael Huemer ~ Stephen Hicks ~ James Henderson ~ Ron Harris ~ Thomas Gramstad Jay Friedenberg ~ John Enright ~ Marsha Familaro Enright ~ Paul Enright ~ Rafael Eilon Peter Chriss ~ Philip Coates ~ Stephen Boydstun ~ Roger E. Bissell

Essays“Rationalism, Skepticism, and Anti-Rationalism

in Greek Philosophy after Aristotle” (V2N4) . . . . . . . . . . . . . . . . . . . . . . . . . . David Potts

“Would Immortality Be Worth It?” (V1N4) . . . . . . . . . . . . . . . . . . . . . . . . . . . . Stephen Hicks

“Can Art Exist without Death?” (V1N5) . . . . . . . . . . . . . . . . . . . . . . . . . Kathleen Touchstone

“The Essence of Art” (V2N5) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Roger E. Bissell

“Evidence of Necessary Existence” (V1N4) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tibor Machan

“Axioms: The Eightfold Way” (V2N2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ronald E. Merrill

“Chaos” (V2N1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Stephen Boydstun

“Capturing Concepts” (V1N1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Stephen Boydstun

“Formation of the Concept of Mind” (V1N6) . . . . . . . . . . . . . . . . . . . . . . . . Paul Vanderveen

“Intricate Consciousness” (V1N5) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Jay Friedenberg

“

Con Molto Sentimento—Music” (V2N3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Marsha Enright“Attentional and Perceptual Disorders

and the Nature of Consciousness” (V2N6) . . . . . . . . . . . . . . . . . . . . . Kathleen Touchstone

“Ascent to Volitional Consciousness” (V1N2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . John Enright

“On the Physical Meaning of Volition” (V1N5) . . . . . . . . . . . . . . . . . . . . . . . Ronald E. Merrill

“Compatibility of Determinism and Free Will” (V2N3) . . . . . . . . . . . . . . . . . . George Lyons

“Volitional Synapses” (V2N1, 2, 4) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Stephen Boydstun

“Beginning—Fulfilling” (V1N3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . James Henderson

“A Philosophy for Living on Earth” (V1N6) . . . . . . . . . . . . . . . . . . . . . . . . . Peter Saint-André

“Epicurus and Rand” (V2N3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ray Shelton

“Objectivist Ethics: A Biological Critique” (V2N5) . . . . . . . . . . . . . . . . . . . . Ronald E. Merrill

“The Subjectivist’s Dilemma” (V2N4) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Michael Huemer

“A Perfectionist-Egoist Theory of the Good” (V2N5) . . . . . . . . . . . . . . . . . . . . Irfan Khawaja

“Ayn Rand: Literary Portraiture v. Philosophic Explication

of Ideal Man and Woman (V2N6) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Charles Wieder

“Imagination and Cognition” (V1N3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Merlin Jetton

“Identity of Indiscernibles and Quantum Physics” (V1N4) . . . . . . . . . . . . . . . . Joseph Mixie

“Induction on Identity” (V1N2, 3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Stephen Boydstun

“Intuition, the Subconscious,

and the Acquisition of Knowledge (V1N6, V2N1) . . . . . . . . . . . Kathleen Touchstone

“The Nature of Numbers” (V1N1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Merlin Jetton

“Philosophy of Mathematics” (V1N2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Merlin Jetton

“Mathematic Empiric” (V1N6) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Daniel Ust

“Mathematics and Intuition” (V2N4) . . . . . . . . . . . . . . . . . . . . . . . . . . . . Kathleen Touchstone

“Reality of Mind” (V2N1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Eyal Mozes

“Why Man Needs Approval” (V1N2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Marsha Enright

“Time, Prescience, and Biology” (V2N2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Merlin Jetton

“On Probability” (V2N1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Merlin Jetton

“Rationality of Decision and Belief” (V2N2) . . . . . . . . . . . . . . . . . . . . . . Kathleen Touchstone

“Pursuing Similarity” (V2N6) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Merlin Jetton

“On Newtonian Relative Space” (V1N6) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fred Seddon

“Space, Rotation, Relativity” (V2N2, 3, 5, 6) . . . . . . . . . . . . . . . . . . . . . . . . . Stephen Boydstun

“Theories of Truth” (V1N4, 5, 6) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Merlin Jetton

“The Turing Test” (V1N5) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Gary McGath

Remarks“Nested Observers and Modeling Requirement” . . . . . . . . . . . . . . . . . . . . . Robert Sweeney

(V1N6 re McGath “Turing”)

“On the Possibility of Artificial Intelligence” . . . . . . . . . . . . . . . . . . . . . . . Thomas Gramstad

(V1N6 re McGath “Turing”)

“Reply to Sweeney and Gramstad” (V1N6) . . . . . . . . . . . . . . . . . . . . . . .Gary McGath

“Wakening” (V1N5 re Henderson “Beginning”) . . . . . . . . . . . . . . . . . . . . . . . . . Philip Wagoner

“Being Who You Are” (V1N6) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Philip Wagoner

“Concerning Consciousness” (V1N2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Marsha Enright

“Determinism and Knowledge” (V1N6 re Merrill “Volition”) . . . . . . . . . . . . . . . . . Ron Harris

“Reply to Harris” (V1N6) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ronald Merrill

“In Defense of Full Physical Determinism” (V2N5 re Boydstun “Synapses”) . . . . Rafael Eilon

“Reply to Eilon” (V2N5) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Stephen Boydstun

“Epicurean Pleasure and the Objectivist Good” . . . . . . . . . . . . . . . . . . . . . Peter Saint-André

(V2N4 re Shelton “Epicurus”)

“Parallel Metaethics” (V2N4 Reply to Saint-André) . . . . . . . . . . . . . . . . . . . Ray Shelton

“On Merrrill ‘Ethics’” (V2N6) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Marsha Enright

“On Merrill ‘Ethics’” (V2N6) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Philip Coates

“Existence Is Independent Individuality” (V2N2) . . . . . . . . . . . . . . . . . . . . . . . . . Peter Chriss

“Relativity of Rotation” (V1N5 re Mixie “Indiscernibles”) . . . . . . . . . . . . . . Christopher Weed

“Mapping Reality” (V2N4) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Stephen Boydstun

“Finitude and Meaning” (V1N5 re Hicks “Immortality”) . . . . . . . . . . . . . . . . James Henderson

“Reply to Henderson” (V1N5) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Stephen Hicks

“Quantity and Number” (V1N2 re Jetton “Numbers”) . . . . . . . . . . . . . . . . . . . . . George Lyons

“The Complexion of Number” (V1N3 re Jetton “Numbers”) . . . . . . . . . . . . . . . . . . David Ross

“From the Ground Up” (V1N5 re Jetton “Numbers”) . . . . . . . . . . . . . . . . . . . . . . . Paul Enright

“True Searcher” (V2N6) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Stephen Boydstun

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## Michael Stuart Kelly 1,653

Stephen,

That's one hell of a story about you and

Objectivity. That's concrete proof that love of productive work is life enhancing.I am interested in receiving the essays. All of them are extremely high-quality. What do I do?

Michael

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## Guyau 11

Michael,

To obtain these articles, visit www.bomis.com/objectivity/ and click on the Prices sector. The prices of any and all issues of

Objectivityare listed there. Instructions for making payment are also given in that sector.When you put your check or money order into the mail, send an e-mail to boydstun@rcn.com. Let us know which issues you have ordered and your US Postal Service address. All issues are mailed Priority.

Thank you for your interest.

Stephen

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## MBM 0

Hey, you've got to enter the 21st century and accept payment via Paypal!

No matter, I'll probably be ordering a bunch if not all of these...many of the titles look very

interesting.

MBM

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## John Dailey 0

Stephen:~ I believe I have some of your beginning issues (2 yrs worth) of

somewhere around (I used to be quite organized, but, as of late...). Been meaning to continue the subscription; definitely all were worth reading.Objectivity~ The question of this thread is probably best answerable from a deeper area: the relationship ('logically') between Induction and Deduction themselves.

~ I've argued elsewhere that accepting Deduction as a useable 'method' (re ANYthing, much less only 'math') requires that one accept Induction as the base for accepting Deduction's continued useability. Some have argued (unclear as to whether they 'deductively' did so, or 'inductively' did so; they never specified, interestingly) against my view, but, I've not really seen what they regard as a 'flaw' in it. --- Given that, as pointed out by some re 'enumerating' items/entities, the relationships Deducible within math are resultant from Inductions from

original observations.~ Interesting thread.

LLAP

J:D

P.S: I do find it unfortunate that LP has had little to extemporize on these subjects' relationships, but for occasional 'aside' commenting. We're talking the very BASE of 'Epistemology', for Pete's sakes!

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## Guyau 11

Hi John,

Your idea that

"accepting Deduction as a useable 'method' (re ANYthing, much less only 'math') requires that one accept Induction as the base for accepting Deduction's continued useability"

reminds me of something I wrote in the 1991

Objectivityessay "Induction on Identity." I have a suggestion for how to test and develope this idea further, but first I'll display the related stretch of my essay.You will recall that in that essay I introduced as an existential principle that I called 'the principle of substantive propagation' and that I took the law of identity (in Rand's conception of it) and the law of noncontradiction to be existential principles as well as formal principles. Here is the passage (V1N3 40):

The modern Humean would continue shamelessly: "Yes, Einstein's field equations, Hamilton's principle, and the conservation of energy may have all held up until now, but what about tomorrow? We concede that since the beginning of time there has been nothing cosmically special about present times per se, but what about present times per se in the future? And again, what about this principle of substantive propagation, this principle that the way things will be grow out of the way things have been? That may have been true yesterday, but what about tomorrow?"I shall give a negative argument, then a positive argument. If the ground of induction is the law of identity and no principle is entirely independent of this law, then my arguments should be ultimately circular---well, perhaps spiral, perhaps widening for the positive argument and narrowing for the negative. At any rate of curvature, we may count our circularity on this issue benign, even virtuous. This is not meant to shield my arguments from other types of rational criticism.The negative argument is this: The principle that the ways things will be grow out of the ways things have been is not a bald analytic statement. It is not a stupid tautology. One could go ahead and say "perhaps it is not always the case that the ways things will be grow out of the ways things have been" without having misunderstood the simple meanings of the terms in the principle. Suppose the principle were false. Would the principle of noncontradiction then be true tomorrow? If the principle of noncontradiction is true tomorrow without having been an identity in time with itself today, then it is a radical, Humean contingent truth. Then, in the Humean mentality, it could be false day after tomorrow. Suppose the principle of noncontradiction be false tomorrow. Then was it true today? Suppose the principle of noncontradiction is true for any tomorrow but only contingently so. Better yet, suppose we just return to reality. There we find all the necessity worth having. There we find the principle of noncontradiction and our principle of substantive propagation always true.[Then comes my positive argument, 41-43]The suggestion I wanted to mention is this. Look into mathematical induction and its relations to logical principles in modern deductive logic. Mathematical induction is essential to all ordered numbers, and these sequences have application not only to unfoldings in time but elsewhere. Mathematics presupposes deductive logic, but then adds its own further postulates. Is mathematical induction (the weak and strong forms of MI and the least number principle follow from one another) one of those additional postulates of mathematics? Or is it part of deductive logic, having uses in logic itself? If it is has useful work to do in deductive logic itself, what specifically?

I think the text by J. L. Bell and M. Machover

A Course in Mathematical Logicmay have the answers to those questions.Stephen

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## Charles R. Anderson 0

Your comments are correct if we limit our interest to the body of knowledge only, but as you point out, the practice of mathematics often involves an integration of math and physics. When the mathematian sets out to develop a new mathematical area for research, he usually wishes to develop a mathematics that has the applicability to the real world that is required for it to have the ability to prove anything. He has to build that in, or at least try to. When that area of mathematical investigation results in some proofs about some modeled aspect of the real world, one still wants to check observations and see if the proof is consistent with the observation. Yes, you can theoretically say the mathematician does only the deductive calculations and reasonings, but if he really does only that, then he may as well be playing chess or some other game. Generally, he draws a salary and has some responsibility to do something useful. Often, he teaches, but why should anyone study math if it has no applications? So, in general mathematians try to develop rules and results that do have some applicability to explaining the world about us. Of course, one could develop rules and apply them mathematically with no interest and no regard for reality and it could be said that you were doing mathematics. But, it would be rather like saying someone was a scientist because they were constantly proposing hypotheses and testing them against reality even though every test showed every hypothesis was false and one was left scratching one's head wondering why anyone would ever even have thought that these hypotheses were worth testing.

So the question is this: Does mathematics contain within its field concern for its usefulness or do we assume that the field should have no responsibility and no interest in its usefulness? Should a university have mathematicians who only play arbitrary deductive games and a physics department which points out when their results chance to be useful and when they are useless? I am willing to allow mathematicians to be concerned with the usefulness of their field of study and even develop proofs of physics in the process. I would also allow the physicists and engineers to use their results and even contribute to the field of mathematics as they wish.

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## Mike Hardy 0

That is greatly exaggerated. SOME mathematicians may usually have in mind

applications that non-mathematicians can use, but often applications are to others

areas of mathematics, which in turn are applied to other areas of mathematics,

etc., and applications to the "real" world may show up 50 years later.

Not true. One CHECKS the correctness of the end result by using only deduction.

How one GETS the end result is another matter. Concept formation, in particular,

is involved, and is an inductive process. And it's an vitually overused cliche to say that

mathematicians rely heavily on "intuition" (although there might not be much agreement

on how to define that term, if anyone were to examine it).

Only if you construe that in the light of my qualifications above. -- Mike Hardy

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