Comments (21)

• Jackson

  Interesting take on it my friend. Well done. Not too tiring or over thought. short and sweet.

  Published: March 17, 2009 11:31 PM

• Deefburger

  nudge nudge Say no more!

  Published: March 18, 2009 9:34 AM

• Gil Guillory

  My math teachers and professors all gave partial credit. But for my engineering professors, it was different: no partial credit. You either get a problem right, or you get it wrong. Now that I'm an engineer, I understand. We don't get partial credit for building a bridge that was *almost* correctly designed. We get sued.

  Published: March 18, 2009 10:09 AM

• pete

  oh i dunno. i'm an engineer too, been working for about 7 years. instead of bridges, i'm at a firm that sells scientific instruments to the private sector, and there are strong pressures to release early, accepting the risk of an immature design in order to start getting $$ sooner. It is understood that if the customer gets screwed by this, the service department will set them right. So the engineers here do get "partial credit".

  The experienced customers know this too, and understand that instruments in the first year or so of a product's life will be buggy, but as scientists, the upside for them is a chance to publish before their more risk-averse peers.

  As for economists... not sure how this translates. But like everyone else, you can't expect them to get it right the first time. In my world, the answer to this is that, if you plan to build 200 of something, don't build all 200 at once, build and sell 10 or 20 at a time, study the ones that break and make each round better than the one before.

  Hopefully the good people steering the ship at the FED or wherever do this too... instead of making a trillion-dollar all-in bet in a game they just walked into.

  Published: March 18, 2009 12:32 PM

• David Spellman

  I also home school my youngest son :). I tell all my children that mathematics is not like other subjects. Other subjects are subjective and fuzzy. In mathematics, you can always get the correct answer. I don't give partial credit in mathematics, I expect exact answers 100% of the time. Like I say, "Would you like to cross a bridge designed by an engineer who was 90% right on his math?" "Would you like to fly on a plane designed by an engineer who was even 99% correct on his math?" 100% is the only passing grade.

  Published: March 18, 2009 12:40 PM

• gary c stephenson

  i have struggled with the appropriate blend of concepts and details my entire adult life. i have come to believe that the object of the work dictates the degree of initial detail. however, the importance of a strong as train smoke conceptual knowledge must not be ignored. most people in the hard science based professions believe there is no room for optimism when rendering judgements about the quality of outputs, whether the outputs are math problems, bridges, accounting reports, risk assessments,etc. these are conclusions i have drawn because i now understand what it is like to not know and engage in the struggle to know.

  Published: March 18, 2009 2:32 PM

• someone271

  I must say that I find myself in very strong disagreement with your post. Every now and then I come across an article here at mises.org that criticizes the inductive nature of statistical reasoning from experience, as opposed to deductive nature of mathematics (and Austrian economics) via logic from a set of axioms.

  While I agree completely with Mises' critique of logical positivism and more specifically the epistemological problems of economics as a science, I feel there is a consistent and substantial distortion in the representation of mathematics as a field, and how mathematical knowledge is accumulated.

  In my opinion, there are two common fallacies when representing the field of mathematics in mises.org
  1) Mathematics is represented as something that is either right or wrong.
  2) Mathematics is represented as something whose truth is highly self evident.

  "A mathematician who dared argue that one plus one is not two would be laughed out of the field."
  This statement does not only misrepresent mathematics as a field, it is also historically inaccurate. No mathematician would laugh at you for postulating different axioms, as long as you can show that they enable you to gain new insight into mathematical problems.

  The binary base, Boolean Algebra, fields with characteristic 1, are all mathematical models where 1+1 does not equal 2. They have been important to the development of digital communication and computers, and are taught in first year mathematics courses to this day.

  I think many people feel as though mathematical axioms are somehow clear and self evident, and therefore you know what they are and everything is either in accordance or contradiction with them.

  This too is historically inaccurate. Many mathematical postulates where in debate for a long time, and where decided as a matter of consensus among mathematicians, and nothing else! the most prominent example is perhaps the axiom of choice in the field of Set Theory, it was highly shunned and discredited throughout the beginning of the 20th century, but is now accepted as a standard, and those who deny it are considered eccentric (it is interesting to note that the only reason it became part of the mathematical consensus was because so many theorems are contingent on it being true, what blurs the line between inductive and deductive justification in mathematics) other known examples are: intuitionistic logic, and Euclid's parallel postulate.

  The distinction between inductive and deductive knowledge accumulation in mathematics is also much more blurry than how it is portrayed here in mises.org.

  Let's take calculus for example. The justification for the use of calculus when it was first developed by Newton and Leibniz was a positivistic one, based solely on the physical predictions it enabled. The theory itself was a muddled mess based on the inconsistent, ill defined, concept of the "infinitesimal". The field of calculus had to be redeveloped in the 19th by the likes of Cauchy, in order to justify it mathematically, and even that wasn't enough! Some of the atrocities that engineers perform every day in mathematics where shown to be mathematically justified only in the thirties by Abraham Robinson's work on non standard analysis.

  The point of all my rumbling is that mathematics is not clear cut, well-defined field. Its body of knowledge is in many ways subjective, open to revolution or re-evaluation. costumes, notations, standards of proofs, and scientific consensus are just as important in mathematics then in other fields. Many mathematical innovations where accepted on the sole base of of enabling more accurate predictions.

  further reading
  http://en.wikipedia.org/wiki/Axiom_of_choice
  http://en.wikipedia.org/wiki/Intuitionistic_logic
  http://en.wikipedia.org/wiki/Parallel_postulate#Converse_of_Euclid.27s_parallel_postulate

  Published: March 18, 2009 5:17 PM

• Peter

  The binary base, Boolean Algebra, fields with characteristic 1, are all mathematical models where 1+1 does not equal 2.

  You're confusing values with their representation as digit symbols (and operations with their symbols, too). The binary value written "10" has the value 2 -- 1+1 = 2 in binary as in every base; boolean algebra doesn't involve numbers or addition (you might write "+", but it doesn't mean addition). And a "field with characteristic 1" only has a single number - zero. (Addition mod 2 has a 1 and addition, and 1+1=0, though)

  Published: March 18, 2009 7:04 PM

• Jim Fedako

  someone271,

  Sophistry. Pure Sophistry.

  Published: March 18, 2009 8:58 PM

• someone271

  Jim,

  Why does my argument amount to sophistry, I do not think it is a serious reply to wave it away! (and impolite as well)

  Do you deny my account of how mathematical knowledge is accumulated?

  I would love for you to refer specifically to my example of the axiom of choice.

  Peter,
  I do not believe I am confusing the concepts, but that seems to me to be a metaphysical question rather than a mathematical one, regarding the meaning of numbers and equation. (If I'm not mistaken, Kripke wrote extensively about this)
  In a finite field with two elements numbers are, in my opinion, different concepts than in the natural numbers, for example. I agree that the Boolean operator "+" does not represent addition, but rather the "Xor" operator.
  The point is that you can postulate as you wish, even unintuitive things, as long as you can show the results to be meaningful mathematically.

  Do you deny this?

  are there absolute mathematical axioms?

  could you list them?

  Were there no incidents in history where the consensus about specific axioms changes leading to mathematical innovation? (like hyperbolic geometry)

  Are you a platonist in regards to philosophy of mathematics? (just curious)

  P.S.
  sorry about the error about the characteristic of the field, I meant characteristic 2

  Published: March 19, 2009 12:45 AM

• Brian Macker

  Someone271,

  You state: "Every now and then I come across an article here at mises.org that criticizes the inductive nature of statistical reasoning from experience, as opposed to deductive nature of mathematics (and Austrian economics) via logic from a set of axioms."

  In 1987 Popper and Miller published with the Royal Society (followup to a 1983 proof in Nature Magazine) titled "Why Probablistic Support is Not Inductive".
  

  This paper attempts to provide both an elaboration and a strengthening of the thesis of Popper Miller (Nature, Lond. 302, 687f. (1983)) that probabilistic support is not inductive support. Although evidence may raise the probability of a hypothesis above the value it achieves on background knowledge alone, every such increase in probability has to be attributed entirely to the deductive connections that exist between the hypothesis and the evidence.

  I read it a long time ago and it appeared valid to me.

  There have been attempts to refute this of the kind that say "If I have a theory that a penny is fixed then as I flip it more and more the proportion of tails remains higher than 50% then obviously increases the probability that I have a bad penny". Unfortunately this is not true, since it is possible that some other factor, some other theory is the cause. For example the table may have some kind of magnet that effects pennies that way.

  It fails for the same reason that seeing more and more white swans does not mean it is due to your theory that all swans are white. It could just be that the correct theory behind the selection of only white swans is that you don't live in an area with black ones.

  Published: March 19, 2009 1:49 AM

• Brian Macker

  In case I wasn't clear I was disputing your assumption that statistical reasoning was truly inductive in nature as per "the inductive nature of statistical reasoning from experience".

  Science isn't inductive. Proper science is about pan-critical rationalism.

  Published: March 19, 2009 1:53 AM

• Brian Macker

  Someone271,

  ... and BTW you were thinking of abstract algerbra's rings and groups, not binary arithmetic. I believe it is a integer group modulo 2.

  If I remember correctly from 30 years ago. In a group modulo 2 there are only two values 0 and 1. The binary addition operation is defined as (a + b) mod 2. Thus 1+1 = (1+1) mod 2 = 2 mod 2 = 0, because it wraps around modulo 2. The reason it doesn't equal 2 is because the group doesn't have a 2.

  Published: March 19, 2009 2:07 AM

• someone271

  Brian,

  Thanks a lot for the paper. I'll try and see if I download the entire thing from my university library.

  You are of course correct, in a field with two elements 2 does not exist, what I'm trying to point is that it is as valid a definition of a number as any other. Concepts are generally hard to define, numbers included. I do not think that there is a "true" mathematical definition of a number. That's why I think that in mathematics all axioms are valid as long as they enable mathematical insight into questions.

  I also tend to agree that the proper paradigm for science is pan-critical rationalism, but I have my doubts whether this is actually the case in reality.
  (Thomas Kuhn's "The Structure of Scientific Revolutions" jumps to mind)

  When observed under scrutiny a lot of mathematical knowledge gathered over the past several hundred years was greatly affected by trends, fashions, and the further results it enabled. Concepts like Cardinality of sets changed dramatically only because they proved so useful over time. I'm not sure how you would justify the use of a certain axiom, when your only motivation is that it's useful. (any ideas?)

  Thanks again for the serious answer, it is much appreciated.

  Published: March 19, 2009 5:41 AM

• Brian Macker

  Kuhn was proven wrong. His theory was falsified. He is popular in the English and social sciences departments. Below something I had posted elsewhere.

  I was poking around the Friesian web site when I came across this article that jogged my memory. The topic is criticism of Poppers philosophy of science, and in particular falsification.

  To put some context to the first sentence please remember that Popper himself had already assumed and explicitly addressed the issue of "explaining away falsifying evidence".

  Here we have the interesting case of an actual event that falsifies Kuhn's theory of science. I'd forgotten about it.

  "Kuhn's critique contains a considerable element of truth, since people, scientists included, often do just see what they want to see, and it is possible to explain away falsifying evidence. On the other hand, anomalies are often recognized that are inconvenient to a theory, and most scientists are now aware that if the theory cannot ultimately accommodate them, then a new theory will be necessary. And then there is the Great Devonian Controversy. Once all the major geologists recognized that there was no unconformity in the sequence, and that there were plants in the lower strata, there simply was no existing theory that could accommodate the facts. Thus, we had a situation that, in Kuhnian terms, was impossible. The evidence could not be interpreted to sustain any existing theoretical views.

  The situation was sometimes discussed by participants in what sound like modern terms. Thus, John Phillips, professor of geology at King's College, in London, wrote to De la Beche that his discovery of certain plant fossils, "is a fact to be introduced into the induction, not an anomaly to be frightened at" [p. 223]. Here "anomaly" would mean just what it means now: Some fact inconsistent with a received or desired theory. As "Baconians," both Phillips and De la Beche think that the appropriate theories will be produced by "induction" from a sufficient catalogue of facts. Neither Kuhn nor Popper could agree with that old "Baconian" view of theories, but Kuhn and deconstructionists cannot allow that an anomaly all by itself could potentially falsify a theory, rather than vice versa. But in the Devonian Controversy, the anomalies overthrew all the theories."

  
  

  Published: March 19, 2009 7:35 AM

• Brian Macker

  271, BTW I got your main point and don't disagree.

  Published: March 19, 2009 7:36 AM

• Jim Fedako

  someone271,

  Impolite? Maybe. But you switched concepts. Boolean algebra deals with logic gates, on-off switches, etc. It does not deal with integers or real object.

  So a boolean 1 is not the same as the concept "one" as I used it.

  In addition, your argument about binary numbers is also off. While 1 plus 1 in base two is represented as 10, it is still "two."

  I call such arguments sophistry.

  Where we do agree is with these statements in your last paragraph: "The point of all my rumbling is that mathematics is not clear cut, well-defined field. Its body of knowledge is in many ways subjective, open to revolution or re-evaluation."

  Can't remember who said this, but I agree: 'The only mathematical proofs that are nonrefutable are those that are trivial.' (or something along those lines)

  Published: March 19, 2009 11:47 AM

• someone271

  Brian,

  Thanks for the article, it proved insightful.

  To set the record straight, I consider myself a Popper fan, I do not subscribe to "Social constructionism" , nor am I a big Kuhn fan. I do however believe there is merit in considering how social aspects of science affect it.

  I think that in many ways this draws the line between how science should be conducted, and how it is conducted in practice (which is an entirely different story)

  Thanks again for all the help

  Published: March 19, 2009 11:55 AM

• someone271

  Jim,
  I may be a fool, I may be an undergrad who doesn't really understand mathematics. If that is the case you could in a polite and articulate manner show me my erroneous ways. Instead you chose to offend my intellectual integrity, which I found to be completely unwarranted and offensive.

  To my actual point. You speak as though there is a clear and definitive definition to the concept of "number" or "2", I tend to disagree. a Boolean 1 is not "one" as you used it, but why is it any less valid as a definition?

  Why is your definition better than any other?

  I still fail to see why would anyone be "laughed out of the field" for disputing certain axioms, as long as you can show your axioms to be useful.

  Published: March 19, 2009 12:48 PM

• Jim Fedako

  My apologies. But remember, this is the internet, and you are posting anonymously, so I do not know anything about you. I was just reacting to what appeared – to me, anyway – as sophistry.

  I could argue that you do not exist – at least from my point of view. For all I know, you are some artificial intelligence posting comments. Given that assumption, how could I – in my view, anyway – ever offend you?

  You and I are able communicate because we accept – for the most part – definitions of words, along with other conventions of communication. We can always drift to the metaphysical and deconstruct language to the absurd. But, even here, we will be accepting an underlying meaning despite the words used – the reason that Deconstructionism breaks down in the end.

  That said, if you want to propose to your math professor that the integer 1 plus the integer 1 is something other than the integer 2, and that simple arithmetic is therefore suspect, have at it.
  

  Published: March 19, 2009 2:31 PM

• (8?»

  As a home-schooling parent, I wonder why education has any need to be "scored" to begin with. All it is good for is statistical analysis in the form of a flawed measuring stick. I don't see any more validity in educational statistics than Mises saw in economic ones, as there is nothing quantifiable about the process of human understanding. Charting hits and misses distracts from the purpose of education (learning), while redirecting focus to testing well (or as I like to say, understanding the coherence/beliefs/frame of reference of the test writer (making all tests inherently an exercise in social studies, which btw isn't appreciated by said writers!)).

  I've countless hours of graded education throughout my professional student life. These hours though, pale in comparison to the ungraded periods where learning is done merely in order to advance my own understanding.

  Other than playing along with mainstream educators, I cannot fathom the benefits structured testing/grading allegedly provide. Worse still, is getting my child over the stress this atmosphere has created in some subjects.

  The goal of testing isn't so much to measure understanding as it is to allow the opportunity for the student to demonstrate it. Numbered grades, requiring discussion to determine whether or not "partial credit is applicable" only serve to confuse these two different goals.

  Published: March 19, 2009 6:24 PM