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Why is the objectivity of Mathematics an important (& ID-relevant) question?

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Blind men and an elephant. Notice, the mental conceptions respond to an actual reality and a sufficient experience will provide an objective picture.

In recent days, I have taken time to show that while subjects study the logic of structure and quantity (= Mathematics, in a nutshell), the body of knowledge — including axiomatised systems — is objective. Where, “objective” effectively means, tied to such a body of accountable warrant and to foundational self-evident facts that the substance of that body of knowledge is credibly an accurate description of facets of reality, as opposed to being dubious (though not necessarily false) figments of a subject’s imagination. Of course, while objectivity implies credible truth (truth being the accurate description of relevant reality) it cannot guarantee utter freedom from error or gaps; especially after Godel’s key incompleteness results.

Why is that?

For one, it has been challenged, in a way that currently admits of some objectivity but highlights that some subjectivity attaches, especially to axiomatisation. This is a subtle issue, as in our day and age there would be a rhetorical effect by which the subjective would effectively eat up the objective in the minds of all too many. And, obviously, many features of Mathematics reflect conventions, such as that we routinely use place value, decimal notations, the symbols and functions we use or emphasise are chosen similarly (there are usualy credible alternatives available, e.g. up to the 1600’s Astronomical calculations were routinely based on 60, hence the number of minutes in an hour), etc. So, the general notion that Mathematics beyond certain practically useful but theoretically minor results [like 2 + 2 = 4] is essentially subjective, could easily be promoted.

Were such a notion to be spread, it would taint disciplines that rely on Mathematics directly, it would bring to bear the “how much more” inference for other subjects, and it would bring logic and rationality itself under question. And once such are put in the dock, in an era where the selectively hyperskeptical mentality is king, the door would be open for manipulation of the public into an undue cynicism about responsible and credible but inconvenient bodies of knowledge. Typically, by projecting the accusation or suggestion of oppressive imposition and the like — the classic cultural marxist agit-prop tactic. Which, for example, is the underlying insinuation in the slander that ID is little more than Creationism in a cheap tuxedo.

In short, it is important for us to defend Mathematics as a key part of the hard core of objectivity of knowledge, and especially contemplative, structured, systematised, abstract knowledge.

How can such be done?

First, by laying out a sound and reasonably accessible summary of how the relevant objectivity comes to be, with highlighted key case studies, answering the popularisers of relativism and/or subjectivism about Mathematics (and onward other key disciplines). For example, here is my rough, first draft in a current thread on the onward discussion:

The surreals, in a sketch map of numbers great and small. The fact that there are several approaches to surveying the domain of numbers which converge on a common core while also highlighting other unique but connected facets, shows the existence of a hard factual core.

KF, 12: >>You will observe, I start with math facts, such as of course the existence of the natural counting numbers from zero and their endlessness [cf, app A below], which implies a new class of quantities, the transfinite. Additive inverses also arise as math facts (originally seen from Accounting and the meaning of money owed) [–> thus, negative numbers]. Fractions arise from sharing or parts and wholes, and ratios allow representation. Mixed numbers then come in as wholes and parts. We then can standardise on fractional powers of ten (as the main case) and define place value notation decimal numbers. These turn out to be compressed power series. We have gone to rational numbers.

The big bridge is the irrationals and the continuum, which was seen in antiquity. The side and diagonal of the square are incommensurate [ –> as per Pythagoras, d^2 = s^2 + s^2, so d = sqrt(2) * s, where it can be shown that if this is assumed rational, sqrt(2) * s = p/q, p & q being whole numbers, odd numbers would be even, i.e. we have a reduction to absurdity and know sqrt(2)  is not a rational number, by the logic of structure and quantity]. And, we are looking at a bridge to the new province, Geometry. Going forward we can use trigonometry and co-ordinate Geometry to bridge the arithmetic and the Geometric. Then also, symbolising and variables gets us to our first Algebra [there are many Algebras such as Boolean and Matrix etc].

Complex numbers viewed as rotations allow us to bring space within the ambit so far. Vectors on ijk as orthogonal units — I skip Quaternions — allow us to factor in 3-d space, time gives us the fourth dimension. Vectors, Matrices and Tensors follow as framing new domains of structure and quantity.

Of course, from variables we go to relationships, mappings and functions. Calculus comes in as we look at rates and accumulations of change in space, time, value etc. The concept of a physical measurement as an extended ratio to a standard amount of a quantity allows us to represent scaled phenomena using techniques of Coordinate Geometry and to access Calculus.

All of this is before axiomatisation, set theory and the like.

When the non-Euclidean Geometry breakthrough happens, and axiomatisation is gradually generalised, standardised and established as gold standard, it does so i/l/o a thousands of years old cumulative body of facts, phenomena, reliable methods and more. That axiomatisation and non Euclidean Geometries then feed back into Physics with the General Theory of Relativity.

I suggest, that were axiomatisations put on the table that were not compatible with the body of established facts and knowledge, they would not have been taken seriously.

So, again, we see how objectivity pervades the discipline, including when axiomatisation enters as a means of wide but post Godel, not universal, unification and coherence.

[Let me add, from Nesher, an illustration that seeks to summarise Godel on epistemology of Math:]

Nesher’s summary of Godel on epistemology of Math. Note, under C there is a gap between what proofs access on a coherent axiomatisation and actually accurate facts on mathematical reality. Notice also rational intuition as a primitive of Mathematical contemplation.

Mathematics is not arbitrary or a mere matter of personal unconstrained choices.

Of course, such axiomatisation also ties in to the possible worlds frame, as we see that we are exploring abstract logic model worlds of possibilities. Arguably, they have a real albeit abstract existence.

Indeed, through the significance of distinct identity, we see how numbers become part of the framework for any possible world, and also how logic is inextricably entangled in both mathematics and in an actualised physical world.

That is a powerful result.

All of this ties into logic of being also, hence an exploration of wider reality through possible worlds analysis.

All of this gives us reason to value and prudently use the power of thought.

And along the way, we can see a reason why the summary that Mathematics is the logic of structure and quantity is also credibly significant.>>

Now, is this easy? No [especially as it calls for significant effort to develop familiarity and confidence on a subject many will shy away from . . . as is familiar from the ID debates], but it is where we have to go, to counter the tidal wave of radically relativist narratives that are spreading across our civilisation, tainting as they go.

Fair comment: the rot is deeper rooted and more widely spread than we may at first think.

So, we must awaken to our peril now, and stand for objectivity. END

PS: A survey of key, core facts and connexions in the study of the logic of structure and quantity, AKA, Mathematics:

>>I am thinking, let me start with numbers per the von Neumann construction and show how the core quantitative structure emerges through logical connectivity and then extends without upper limit, implying w as order type of the natural succession of counting numbers, :

The set that collects nothing is {}

Now, we assign:

{} –> 0

However, this is now a distinct albeit abstract entity (and one that exists in any possible world) so:

{0} –> 1

We can now continue collecting, where the RHS of the arrow is a numeral, a name for a number which designates the order type of the set on the LHS:

{0,1} –> 2

{0, 1, 2} –> 3

. . .

{0, 1, 2 . . . k} –> k+1

. . .

That is, without limit.

We may now recognise a new type of quantity, limitless countable succession, the first transfinite ordinal:

{0, 1, 2, . . . k, . . . } –> w [omega]

Omega has cardinality aleph-null.

By using a different and richer strategy, we may lay out the surreal numbers [a representation of which I will now add to the OP], which allows us to extend to numbers that are partly whole and partly fractional, capturing first the rationals then by extension the continuum so the reals. along the way we incorporate negatives as additive inverses a + (-a) = 0. In simple terms if you owe $a and pay $a you clear a debt, you do not have a positive value.

Also, we may freely extend the transfinites and speak to hyperreals which involve infinitesimals, numbers arbitrarily close to 0 in the first instance and which can be seen as reciprocals of numbers that exceed any finite real value. Such form a cloud that can by addition surround any specific real number.

The complex come in as disguised vectors, where we suggest a rotation such that i*a is orthogonal to the real line, then that i*i*a is -a, so that i*i = -1. This opens up a powerful onward world. Including, that we can define angles in the plane. A three dimensional extension opens up models of 3-d space, and concepts of temporal succession and inertia open up model worlds that can map to the physical one we experience.

So now, we have a system of numbers and open the way to further structures such as vectors [and phasors, rotating vectors], matrices, power series, functions, operations [including of course Calculus] and transformations and much more.

The above process is accountable, logically connected, opens up successive logical model worlds and shows true connexions.

Let me note a key logical property of deductive chains, the weakest link principle:

a => b => . . . f, where f is false or self contradictory shatters the chain. We then have to find the key failure point and fix it. And yes, this is used heavily in the reductio ad absurdum proof technique. Systems are only accepted into the fold if they stand up to this.

Where also, let us recall the double-edged sword of implication: a true proposition will only properly imply a true consequent that it is a sufficient condition for, but a false antecedent indiscriminately implies true and false consequents. And also, implication is not equivalence, that requires double implication, often represented by IFF for if and only if.

(Two linked fallacies are affirming the consequent and denying the antecedent. Just because p => q and q is so does not mean p is so unless q also implies p. Likewise, if p => q and p is false does not entail that q must be false as some true r may be such that r => q. And in any case, q’s truth is a matter of accurate description of some relevant reality, which can be logical, relational, structural or quantitative, not just physical. I add: reality is the state of affairs that exists across actualised and abstract worlds, truth accurately describes some targetted facet of reality to which it refers. Truth says of what is that it is, and of what is not that it is not.)

On these and similar factors, the overall system, axiomatisation and all, holds objective truth. Truth on the logic of model worlds, tied back one way or another to historic core schemes, principles and facts tied to clear realities.

Then now, we have in effect a paradigm, which we can extend to connected model worlds that use sets of postulated start-points, axioms and are used to elaborate systems of thought that need not connect to physically observed realities (but often turn out to be surprisingly relevant). Such systems however will cohere through requisites of inner self-consistency and connexions to the core model worlds, force of logic applied to structure and quantity and patterns set by paradigms.

This is the context in which objective truth first speaks to the accountable logical ties, then also to the connexions into the core of established facts and systems. Much of which actually antedates and is materially independent of axiomatisation schemes — we accept certain key axiomatisations in the first instance because they sufficiently comprehend significant domains and give them credible though not certain — Godel counts here — coherence; though we obviously face undecidables and the premise that a system that addresses a complex enough domain comparable to Arithmetic and captures all true statements will be incoherent. (Actually, existence of truths unreachable by axiomatisation schemes that are coherent and perforce limited actually substantiates independent reality, that we are looking at real albeit abstract entities! For, these are obviously not whims and fancies tossed up by our fevered imaginations.)

We have objectivity and warranted, credible reliable truth as an overall pattern in the system. Where, we may fork possible model worlds such as Euclidean/ non Euclidean, or even ZF + C or ZF with something other than C etc.

Mathematics exponentiates its power through its complex, coherent interconnectivity AND its power to lay out and explore abstract, logical model worlds.

(And BTW, I suspect we are here seeing some of the roots of trouble with the idea of fine tuning of the observed cosmos, as a lot of that is explored through what if sensitivity analysis of model worlds initially developed by exploring the dynamics of the observed cosmos.)>>

30 Replies to “Why is the objectivity of Mathematics an important (& ID-relevant) question?

  1. 1
    kairosfocus says:

    Why is the objectivity of Mathematics an important (& ID-relevant) question?

  2. 2
    News says:

    kairosfocus, this (“So, the general notion that Mathematics beyond certain practically useful but theoretically minor results [like 2 + 2 = 4] is essentially subjective, could easily be promoted. Were such a notion to be spread, it would taint disciplines that rely on Mathematics directly, it would bring to bear the “how much more” inference for other subjects, and it would bring logic and rationality itself under question”) is actually happening now in science education in North America.

    It is the latest move by utterly corrupt and incompetent admin/union chokeholders on compulsory education systems. They can’t teach (and don’t want to), so they are making sure that there is nothing to teach. That makes their failure easy to conceal.

    Students in failing schools will probably learn most of what they know from the electronic devices surrounding them, which operate in a “facts non-optional” environment and require at least basic literacy and numeracy from the user.

    Watch for an article from me on this subject.

  3. 3
    kairosfocus says:

    And, in case you think this is not real as an issue, Quora, first several responses:

    https://www.quora.com/Is-mathematics-subjective

    >>Is mathematics subjective?
    10 Answers
    Brian Bi
    Brian Bi, uses math
    Answered Apr 6, 2018 · Upvoted by Patrick Tam, PhD Mathematics, University of California, Davis (2018) and Bharathi Ramana Joshi, have been doing maths for over 10 years now · Author has 3.7k answers and 30m answer views

    If we define “mathematics” as a human activity that mathematicians engage in, then yes, certain aspects of mathematics are subjective.

    [–> notice the game implicit in “human activity” definitions. Here, while there are different, by and large patently convergent views, I go with a mod from my M100 prof: Mathematics is the logic of structure and quantity, where of course it is studied by capable investigators who share accountable findings, cumulatively building a body of knowledge and good practice]

    For example, mathematicians have subjective opinions about whether the axiom of choice “should” be accepted, and, more generally, what axioms and forms of logical deduction are acceptable. The ongoing controversy over Shinichi Mochizuki’s Inter-Universal Teichmüller Theory and his alleged proof of the abc conjecture illustrates that mathematicians can also have subjective opinions about whether a conjecture has really been proven (if the author has not communicated their ideas clearly enough, then it has not really been proven, but people disagree on whether this has happened).

    [–> it’s controversial, therefore subjective does not work. Only failure or absence of accountable warrant achieves this. Think about the implications of the there is disagreement therefore it’s subjective standard for the court-room and of course for the media. Yes, the undermining of the significance of warrant on credible facts and logic is opening the door to the nihilism of might and manipulation make ‘right’/’truth’/ ‘knowledge’/ ‘justice’/ ‘rights’ and more. Thence, absurdity and chaos.]

    There is also subjectivity in how the value of mathematical work is assigned. Some proofs are more elegant than others. Some research does more to advance mathematics than others, and those who do the “worthiest” work receive faculty positions, research prizes, and so on, but there is no algorithm to determine the value of work in this way.
    2.3k Views · View Upvoters · Answer requested by Jishnu
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    Christopher Mackinga
    Christopher Mackinga, Software Engineer (2017-present)
    Answered Apr 6, 2018 · Author has 64 answers and 6.3k answer views
    Originally Answered: Is there subjectivity in mathematics?

    This is a tough one to describe eloquently. I would say that mathematics is humanity’s subjective model of objectivity. I would also say it’s the closest we’ve gotten to creating or describing something objectively. What I mean by this is that although it’s not objective in itself, it is still objective to each individual. Mathematics is an abstract subjective representation of things which objectively exist. 1 + 1 = 2 not because it’s an objective fact, but because we all agree that this abstraction accurately represents something which objectively exists.

    I’m not sure I’ve done a good job of explaining my position, but if I was to try and summarize my answer to your question, it would go something like this;

    Mathematics is completely subjective, but it represents things which are completely objective.

    Math can be said to objectively exist as an abstraction within our minds, but math without the things it describes/represents is mostly meaningless. (Yes I mean subjectively meaningless. Objectively it is completely meaningless as meaning is a subjective construct)
    567 Views · View Upvoters · Answer requested by David Moore
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    Dane Dormio
    Dane Dormio, Online Tutor & Academic Coach
    Answered Dec 10, 2016 · Author has 75 answers and 34.6k answer views

    I would say it depends on which definition of “subjective” you are referring to. For example, the following definitions come from Dictionary.com: the definition of subjective

    “existing in the mind; belonging to the thinking subject rather than to the object of thought”
    pertaining to the subject or substance in which attributes inhere; essential.

    I think in this sense, you could say that math is subjective, in that it exists “in the mind”. I once read a math book that conveyed what I believe is a related sentiment: that mathematical objects are precisely defined, whereas physical objects are somewhat “fuzzy”, and in this sense, mathematics is “concrete”, whereas the physical world is “abstract”!

    However, the more common meaning of “subjective” is more along the lines of these definitions:

    pertaining to or characteristic of an individual; personal; individual:
    placing excessive emphasis on one’s own moods, attitudes, opinions, etc.; unduly egocentric.
    relating to properties or specific conditions of the mind as distinguished from general or universal experience.

    In this sense, mathematical truths, by virtue of being self-evident and independently verifiable, are not subjective but objective.
    773 Views · View Upvoters>>

    The next response is interesting also:

    >>Francesco Iovine
    Francesco Iovine, No numbers no life
    Answered Apr 6, 2018 · Author has 1.9k answers and 1.3m answer views
    Originally Answered: Is there subjectivity in mathematics?

    Only during the discovery process. Once a theorem is demonstrated, either somebody finds that there is some flaw in the demonstration, or it is objectively accepted.

    There is indeed an approach to mathematics called Constructivism (mathematics) – Wikipedia that requires some strictier logic rules with respect to what is normally accepted. For instance you cannot demonstrate the existence of something by demonstrating that it is impossible that it does not exist.

    This is somewhat a form of subjectivity, as these mathematicians make a strong statement about what a demonstration should be that other mathematicians do not agree with.
    275 Views · View Upvoters>>

    This question is on axiomatisation:

    https://www.quora.com/Are-mathematical-axioms-subjective-and-do-they-make-math-subjective

    >>Are mathematical axioms subjective, and do they make math subjective?
    3 Answers
    David Joyce
    David Joyce, Ph.D. Mathematics, University of Pennsylvania (1979)
    Updated May 8, 2017 · Upvoted by Vinay Madhusudanan and James Brust · Author has 4.7k answers and 15.4m answer views

    Axioms are definitions. They define the subject under investigation.

    If you want to investigate how something works when it’s got two operations like addition and multiplication, then you need to state precisely what properties those operations satisfy. If you include enough of them, you’ll get the axioms for fields.

    If you want to investigate plane geometry, you have to specify exactly what properties plane geometry satisfies. If you use Euclid’s axioms, you get Euclidean geometry. If you modify one the axioms, the parallel postulate, you’ll get a different geometry, called hyperbolic geometry. Both kinds of geometries are part of mathematics.

    The axioms themselves are not subjective. Neither is mathematics as a whole.

    What interests people will determine what parts of mathematics are investigated. That could be called subjective. In large part, the development of mathematics has been influenced by other subjects. Much of mathematics was developed for business, astronomy, engineering, government, physics, and other human endeavors.
    713 Views · View Upvoters
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    Daniil Kozhemiachenko (?????? ???????????)
    Daniil Kozhemiachenko (?????? ???????????), student of complexity theory and symbolic logic
    Answered May 2, 2017 · Author has 741 answers and 705.9k answer views

    They pretty much are. For example, there are still some mathematicians who don’t accept axiom of choice. Even here, on Quora (@David Joyce for example, if I am not mistaken).

    There are even some mathematicians that reject classical logic at all, preferring intuitionistic in its stead. There is even an opinion that for any logic there can possibly be a mathematic (more precisely, a set theory). Hence there are some results accepted by one mathematicians but rejected by others.

    So, one can freely choose any extension or restriction of any set theory one wants provided that this modification does not “explode” (i.e. does not entail any statement: in many non-classical logics it is not the same as “inconsistent”). Does it make math subjective? Maybe. Does it change anything substantial? I doubt.
    153 Views · View Upvoters
    Yasa Herati
    Yasa Herati
    Answered May 4, 2017 · Author has 257 answers and 184.1k answer views

    The best part about math is its lack of subjectivity. Yes, there are different ways to get to an answer, but at the end of the day, the answers are not subjective. Questions like P=NP may have opinions for now, but one day they may be proven. Then, an objective answer will be discovered. Mathematical axioms are not subjective, and they do not make math subjective.
    94 Views · View Upvoters · Answer requested by Dan Tolov>>

    In short, the issue is real and indeed opens up the how much more argument.

    KF

  4. 4
    kairosfocus says:

    News, what a mess we are in! As a civilisation carelessly, stubbornly walking on the crumbling edge of a cliff. KF

  5. 5
    bornagain77 says:

    Of related note, objective reality does not belong to the Atheistic Materialist but to the Theist who holds ‘subjective’ Mind to be the primary foundation of reality:

    On The Comparison Of Quantum and Relativity Theories – Sachs – 1986
    Excerpt: quantum theory entails an irreducible subjective element in its conceptual basis. In contrast, the theory of relativity when fully exploited, is based on a totally objective view.
    http://books.google.com/books?.....38;f=false

    How (conscious) observation is inextricably bound to measurement in quantum mechanics:
    Quote: “We wish to measure a temperature.,,,
    But in any case, no matter how far we calculate — to the mercury vessel, to the scale of the thermometer, to the retina, or into the brain, at some time we must say: and this is perceived by the observer. That is, we must always divide the world into two parts, the one being the observed system, the other the observer.”
    John von Neumann – 1903-1957 – The Mathematical Foundations of Quantum Mechanics, pp.418-21 – 1955

    Steven Weinberg, an atheist, rightly rejects the ‘realist approach’ to quantum mechanics mainly because of ‘many worlds interpretation’, but, on the other hand, it is interesting to note the main reason for why he, as an atheist, rejects the ‘instrumentalist approach’ to quantum mechanics:

    The Trouble with Quantum Mechanics – Steven Weinberg
    – January 19, 2017
    Excerpt: The instrumentalist approach,, (the) wave function,, is merely an instrument that provides predictions of the probabilities of various outcomes when measurements are made.,,
    In the instrumentalist approach,,, humans are brought into the laws of nature at the most fundamental level. According to Eugene Wigner, a pioneer of quantum mechanics, “it was not possible to formulate the laws of quantum mechanics in a fully consistent way without reference to the consciousness.”11
    Thus the instrumentalist approach turns its back on a vision that became possible after Darwin, of a world governed by impersonal physical laws that control human behavior along with everything else. It is not that we object to thinking about humans. Rather, we want to understand the relation of humans to nature, not just assuming the character of this relation by incorporating it in what we suppose are nature’s fundamental laws, but rather by deduction from laws that make no explicit reference to humans. We may in the end have to give up this goal,,,
    Some physicists who adopt an instrumentalist approach argue that the probabilities we infer from the wave function are objective probabilities, independent of whether humans are making a measurement. I don’t find this tenable. In quantum mechanics these probabilities do not exist until people choose what to measure, such as the spin in one or another direction. Unlike the case of classical physics, a choice must be made,,,
    http://www.nybooks.com/article.....mechanics/

    Since Weinberg rejects the instrumentalist approach to quantum mechanics since it undermines Darwin’s goal of trying to explain humans as purely the result of the laws of nature, if I had the pleasure, I would like to ask Weinberg exactly who discovered the standard model? Weinberg or the laws of nature?

    Also of note to the ‘realist’ approach in quantum mechanics, “realism” has now been falsified:

    Realism in the sense used in physics[6] is the idea that nature exists independently of man’s mind:
    – per wikipedia

    An experimental test of non-local realism – 2007
    Simon Gröblacher, Tomasz Paterek, Rainer Kaltenbaek, Caslav Brukner, Marek Zukowski, Markus Aspelmeyer & Anton Zeilinger
    Abstract: Most working scientists hold fast to the concept of ‘realism’—a viewpoint according to which an external reality exists independent of observation. But quantum physics has shattered some of our cornerstone beliefs. According to Bell’s theorem, any theory that is based on the joint assumption of realism and locality (meaning that local events cannot be affected by actions in space-like separated regions) is at variance with certain quantum predictions. Experiments with entangled pairs of particles have amply confirmed these quantum predictions, thus rendering local realistic theories untenable. Maintaining realism as a fundamental concept would therefore necessitate the introduction of ‘spooky’ actions that defy locality. Here we show by both theory and experiment that a broad and rather reasonable class of such non-local realistic theories is incompatible with experimentally observable quantum correlations. In the experiment, we measure previously untested correlations between two entangled photons, and show that these correlations violate an inequality proposed by Leggett for non-local realistic theories. Our result suggests that giving up the concept of locality is not sufficient to be consistent with quantum experiments, unless certain intuitive features of realism are abandoned.
    http://www.nature.com/nature/j.....05677.html

    Quantum physics says goodbye to reality – Apr 20, 2007
    Excerpt: Many realizations of the thought experiment have indeed verified the violation of Bell’s inequality. These have ruled out all hidden-variables theories based on joint assumptions of realism, meaning that reality exists when we are not observing it; and locality, meaning that separated events cannot influence one another instantaneously. But a violation of Bell’s inequality does not tell specifically which assumption – realism, locality or both – is discordant with quantum mechanics.
    Markus Aspelmeyer, Anton Zeilinger and colleagues from the University of Vienna, however, have now shown that realism is more of a problem than locality in the quantum world. They devised an experiment that violates a different inequality proposed by physicist Anthony Leggett in 2003 that relies only on realism, and relaxes the reliance on locality. To do this, rather than taking measurements along just one plane of polarization, the Austrian team took measurements in additional, perpendicular planes to check for elliptical polarization.
    They found that, just as in the realizations of Bell’s thought experiment, Leggett’s inequality is violated – thus stressing the quantum-mechanical assertion that reality does not exist when we’re not observing it. “Our study shows that ‘just’ giving up the concept of locality would not be enough to obtain a more complete description of quantum mechanics,” Aspelmeyer told Physics Web. “You would also have to give up certain intuitive features of realism.”
    http://physicsworld.com/cws/article/news/27640

    Should Quantum Anomalies Make Us Rethink Reality?
    Inexplicable lab results may be telling us we’re on the cusp of a new scientific paradigm
    By Bernardo Kastrup on April 19, 2018
    Excerpt: ,, according to the current paradigm, the properties of an object should exist and have definite values even when the object is not being observed: the moon should exist and have whatever weight, shape, size and color it has even when nobody is looking at it. Moreover, a mere act of observation should not change the values of these properties. Operationally, all this is captured in the notion of “non-contextuality”: ,,,
    since Alain Aspect’s seminal experiments in 1981–82, these predictions (of Quantum Mechanics) have been repeatedly confirmed, with potential experimental loopholes closed one by one. 1998 was a particularly fruitful year, with two remarkable experiments performed in Switzerland and Austria. In 2011 and 2015, new experiments again challenged non-contextuality. Commenting on this, physicist Anton Zeilinger has been quoted as saying that “there is no sense in assuming that what we do not measure [that is, observe] about a system has [an independent] reality.” Finally, Dutch researchers successfully performed a test closing all remaining potential loopholes, which was considered by Nature the “toughest test yet.”,,,
    It turns out, however, that some predictions of QM are incompatible with non-contextuality even for a large and important class of non-local theories. Experimental results reported in 2007 and 2010 have confirmed these predictions. To reconcile these results with the current paradigm would require a profoundly counterintuitive redefinition of what we call “objectivity.” And since contemporary culture has come to associate objectivity with reality itself, the science press felt compelled to report on this by pronouncing, “Quantum physics says goodbye to reality.”
    The tension between the anomalies and the current paradigm can only be tolerated by ignoring the anomalies. This has been possible so far because the anomalies are only observed in laboratories. Yet we know that they are there, for their existence has been confirmed beyond reasonable doubt. Therefore, when we believe that we see objects and events outside and independent of mind, we are wrong in at least some essential sense. A new paradigm is needed to accommodate and make sense of the anomalies; one wherein mind itself is understood to be the essence—cognitively but also physically—of what we perceive when we look at the world around ourselves.
    https://blogs.scientificamerican.com/observations/should-quantum-anomalies-make-us-rethink-reality/

    The Death of Materialism – InspiringPhilosophy – (Material reality does not exist without an observer) video
    https://www.youtube.com/watch?v=wM0IKLv7KrE
    Materialism has been dead for decades and recent research only reconfirms this, as this video will show. This video was reviewed by physicist Fred Kuttner and Richard Conn Henry. A few other physicists reviewed this but asked to remain anonymous for privacy reasons.

    Reality doesn’t exist until we measure it, (Delayed Choice) quantum experiment confirms –
    Mind = blown. – FIONA MACDONALD – 1 JUN 2015
    Excerpt: “It proves that measurement is everything. At the quantum level, reality does not exist if you are not looking at it,” lead researcher and physicist Andrew Truscott said in a press release.
    http://www.sciencealert.com/re.....t-confirms

    New Mind-blowing Experiment Confirms That Reality Doesn’t Exist If You Are Not Looking at It – June 3, 2015
    Excerpt: Some particles, such as photons or electrons, can behave both as particles and as waves. Here comes a question of what exactly makes a photon or an electron act either as a particle or a wave. This is what Wheeler’s experiment asks: at what point does an object ‘decide’?
    The results of the Australian scientists’ experiment, which were published in the journal Nature Physics, show that this choice is determined by the way the object is measured, which is in accordance with what quantum theory predicts.
    “It proves that measurement is everything. At the quantum level, reality does not exist if you are not looking at it,” said lead researcher Dr. Andrew Truscott in a press release.,,,
    “The atoms did not travel from A to B. It was only when they were measured at the end of the journey that their wave-like or particle-like behavior was brought into existence,” he said.
    Thus, this experiment adds to the validity of the quantum theory and provides new evidence to the idea that reality doesn’t exist without an observer.
    http://themindunleashed.org/20.....at-it.html

    “The concept of the objective reality of the elementary particles has thus evaporated…”,,,; “The idea of an objective real world whose smallest parts exist objectively in the same sense as stones or trees exist, independently of whether or not we observe them,,, is impossible.,,, We can no longer speak of the behavior of the particle independently of the process of observation”
    – W. Heisenberg, Physics and Philosophy, Harper and Row, New York (1958)

    Albert Einstein vs. Quantum Mechanics and His Own Mind – video
    https://www.youtube.com/watch?v=vxFFtZ301j4

  6. 6
    kairosfocus says:

    BA77, interesting perspective. Today has been busy and the Prince of Darkness is in firm control of our power utility. My thought is, a key matter is that Math deals with abstract logical model worlds that speak to possibilities and actualities. The power of the logic of being then applies and we see that certain things are necessary in any world; e.g. once distinct identity exists then numbers must exist, and from them, their logical implications and constructs upon them. This is part of the Wigner principle on Mathematical effectiveness. Even, the concepts of the quantum world are in large part consequent on that power. And certainly, Observer is inextricably involved in the Quantum world — raising all sorts of issues. Fun stuff comes in also when we look at the hyper-reals, infinitesimals and transfinites catapulted back and forth across the span of the Reals. I guess we can get away with saying complex numbers are vectors manipulated algebraically, using e^i*w = cis w, per power series expansion of the exponential . . . which drags in a transfinite summation and directly implies our old friend 0 = 1 + e^i*pi. That single expression directly implies that to INFINITE exactitude, arithmetic, algebra, geometry and then huge swaths of frequency and transient domain behaviour, analysis (including calculus) and more are locked together in inseparable unity. That in itself is pregnant with implications for the coherence and anchoring of huge fields in math to facts on the ground for centuries and in some cases millennia. Then, the real cruncher: we can only access this huge domain by contemplative, rational, responsible, free reflection. So soon as such is tainted with unconscious non-rational control or blind stochastic patterns, the whole collapses. Now, just what frame of thought is friendly to objectivity as a keystone for Mathematics? KF

  7. 7
  8. 8
    kairosfocus says:

    F/N: This is part of building a base for further action. Ponder the power of the mind to grasp truth, starting with necessary ones, moving on to logical connectivity and coherence of a logical model world and onward power in this actualised, concrete world. Power of mind to grasp abstract but powerful truth, by pure rational, responsible, accountable contemplation — not just by empirical observation and connected inductive inferences. Then we see why Plato’s Academy had a sign over the door that none should come there who was not an effective Geometer. Geometry, of course, being the first axiomatised model world in Mathematics. KF

  9. 9
    kairosfocus says:

    F/N: I have taken time to search various perspectives, and find a representative subjectivist view suitable for a general remark, here:

    >> https://physicalspace.wordpress.com/2008/12/31/mathematical-truths/

    Mathematical truths seem to be absolute, for they seem to be universal and invariable. They seem to be objective, independent from place, culture, age, etc., and they also seem to be eternal. But mathematical truths are in the end subjective and relative.

    First, mathematical truths are subjective simply because mathematics itself is humanly subjective. Mathematical systems are human constructions that don’t have any reality outside the human mind . . . . unlike the dominant belief during the Enlightenment that Nature is a rational order, Nature is non-rational, reason is a human faculty and we use reason to understand Nature often rationalising it. So logical consistency in mathematical truths doesn’t mean universal objectivity.

    The relativity of mathematical truth not only is a necessary consequence of its subjectivity, but it has also some concrete manifestations like Gödel’s incompleteness theorems. The theorems states that mathematical systems (or at least those of any practical interest) include truths that cannot be proven within their system. Furthermore, any proof of their truthfulness would make the system inconsistent, and any attempt to prove those truths from outside the system would involve truths from another system that cannot be proven.

    The consequence of Gödel’s incompleteness theorems is that, we might have a system, and truths within the system, that are logically consistent. But those truths are confined and relative to that system, and there is no way to prove them objectively. Mathematical truths then, are always relative. >>

    Let’s examine on points:

    >>Mathematical truths seem to be absolute, for they seem to be universal and invariable. They seem to be objective, independent from place, culture, age, etc., and they also seem to be eternal.>>

    1 –> The OP points to corollaries of distinct identity, which set up the natural numbers and much more; i/l/o previous OPs in this series; cf. point 12 below also. Such are framework for any possible world — which to be distinct must have particular distinguishing characteristics. (I here appeal to the principle that if what are allegedly two distinct things A and B are in fact identical by way of being utterly indistinguishable, they are the same thing.)

    2 –> This points to logic of being issues, namely that in the core of mathematics, we have a cluster of abstract realities that exhibit the logic of structure and quantity which are necessary, framework beings for any possible world.

    3 –> Such beings will not depend on external, enabling “on/off” causal factors and will have neither beginning nor end. (On the broader picture, if a world now is, SOMETHING that is independent like this always was, as utter non-being . . . the genuine nothing . . . can have no causal power. If ever utter non-being was so, such would forever obtain and there would be no world.)

    4 –> So, deep-rooted logic of being issues lie at the heart of the problem.

    >> But mathematical truths are in the end subjective and relative.

    First, mathematical truths are subjective simply because mathematics itself is humanly subjective. Mathematical systems are human constructions that don’t have any reality outside the human mind . . . . >>

    5 –> Little more than an asserted assumption and perspective-driven definition. In effect, if it comes from our contemplative process and thus from a subject, it must be subjective.

    6 –> At one level, this reflects a failure to mark the difference between what has been subjected successfully to a process of accountable warrant and what simply appears so phenomenologically. That is, warrant is locked out.

    7 –> At the deeper level, this reflects the Kantian ugly gulch, where appearance is utterly separated from the reality of things in themselves and it is inferred that we can only address a locked-in world of appearances. But that is — as F H Bradley long since pointed out — a grossly self-referential and incoherent claim, as one implies to “know” that external reality is un-knowable. It defeats itself.

    8 –> Let’s clip:

    We may agree, perhaps, to understand by metaphysics an attempt to know reality as against mere appearance, or the study of first principles or ultimate truths, or again the effort to comprehend the universe, not simply piecemeal or by fragments, but somehow as a whole [–> i.e. the focus of Metaphysics is critical studies of worldviews] . . . .

    The man who is ready to prove that metaphysical knowledge is wholly impossible . . . himself has, perhaps unknowingly, entered the arena . . . To say the reality is such that our knowledge cannot reach it, is a claim to know reality ; to urge that our knowledge is of a kind which must fail to transcend appearance, itself implies that transcendence. For, if we had no idea of a beyond, we should assuredly not know how to talk about failure or success. And the test, by which we distinguish them, must obviously be some acquaintance with the nature of the goal. Nay, the would-be sceptic, who presses on us the contradictions of our thoughts, himself asserts dogmatically. For these contradictions might be ultimate and absolute truth, if the nature of the reality were not known to be otherwise . . . [such] objections . . . are themselves, however unwillingly, metaphysical views, and . . . a little acquaintance with the subject commonly serves to dispel [them]. [Appearance and Reality, 2nd Edn, 1897 (1916 printing), pp. 1 – 2; INTRODUCTION. At Web Archive.]

    9 –> Now, of course, the issue here is that we are dealing with abstracta, and so the gulch applies with double force. But in fact, precisely because we here deal with especially the core facts of Mathematics, we deal with implications and corollaries of distinct identity. Logic, quantity and structure are framework for any world, and so will affect and constrain what is possible and knowable in the actual one.

    10 –> For specific but instructive instance, let | be a stick (or a finger). It is self-evidently, necessarily true that if we cluster sticks (or fingers — we look at an in-common property) as follows, we will have the result we symbolise as further follows:

    || + ||| –> |||||

    2 + 3 = 5

    That is a two-set joined to a three set will yield a five-set, where if two sets of distinct things can be matched 1:1 they have the same number of elements, the same cardinality . . . which becomes very interesting with transfinites.) And so forth.

    >> unlike the dominant belief during the Enlightenment that Nature is a rational order, Nature is non-rational, reason is a human faculty and we use reason to understand Nature often rationalising it.>>

    12 -> The anonymous source here fails to address the import of distinct identity. This. s/he fails to see that the triple first principles of right reason (and as extended to structure and quantity) necessarily obtain in any distinct world W:

    W = {A | ~A}

    So, IDENTITY: A is itself marked by characteristics that allow distinction.

    Likewise, by partition, NON-CONTRADICTION: no x in W can be both A and ~A under the same sense and circumstances.

    Thus also, EXCLUDED MIDDLE: no y in W will be neither A nor ~A, nor both. A X-OR ~A.

    Where the partition brings with it the distinction between one-ness and two-ness, also the exclusions mark the empty set and define zero. We may apply von Neumann and extend it across the domain of numbers. Or the like, i.e. we see number facts as undeniable realities applicable to any possible world.

    >>So logical consistency in mathematical truths doesn’t mean universal objectivity.>>

    13 –> No one has seriously argued that mere coherence implies objectivity. What is argued is that per effectively self evident first principles of reason that are necessary to a world, we have facts of quantity and structure and a logical process of exploration that provides accountable, reliable — as opposed to utterly infallible — warrant for conclusions.

    14 –> And that is where the objectivity is founded.

    >>The relativity of mathematical truth not only is a necessary consequence of its subjectivity,>>

    15 –> Circularity, without foundation for the assumptions. Errors carried forward, in short.

    >> but it has also some concrete manifestations like Gödel’s incompleteness theorems.>>

    15 –> Assertion, again. In fact, core mathematical facts are antecedent to any particular axiomatisation and as discussed, axiomatic systems are accountable before such, they serve to set up frameworks that allow further systematic, objective inference to warranted onward results. This confers objectivity.

    >> The theorems states that mathematical systems (or at least those of any practical interest) include truths that cannot be proven within their system. Furthermore, any proof of their truthfulness would make the system inconsistent, and any attempt to prove those truths from outside the system would involve truths from another system that cannot be proven.>>

    16 –> Actually, any sufficiently complex system will be such that whatever spans all true claims will be incoherent, thus self-falsifying.

    >>The consequence of Gödel’s incompleteness theorems is that, we might have a system, and truths within the system, that are logically consistent. But those truths are confined and relative to that system, and there is no way to prove them objectively. Mathematical truths then, are always relative.>>

    17 –> Unwarranted inference. So long as there are antecedent facts of Mathematics forming a centuries old body of knowledge back to Euclid, Pythagoras and beyond [recall the irrationals were discovered what 2500 years ago], mathematical systems are accountable before facts of mathematics and no axiomatisation grossly out of line would be reasonable. This confers objectivity.

    18 –> What Godel actually showed is that the logic of axiomatisation implies undecidables, things that are true of a domain [which is obviously antecedent to the axiomatisation] which are true independent of the scheme of axioms.

    19 –> So, this actually points to the objectivity.

    20 –> But we live in a day and age besotted with relativism and subjectivism.

    KF

  10. 10
  11. 11
    Eugene S says:

    Mathematics is a language used for describing reality. The first is a map and the second is the territory. Is a map subjective or objective? If it is detailed enough, we can move about confidently, using the map.

    However, some level of subjectivity does come into play because science itself necessarily involves cognitive bias in how we introduce the epistemic cut between the object and the subject, the observed and the observer, the known and the knower. Scientific analysis requires counterpart synthesis. We are part of this world and when we study it rationally, we cannot get away from the question of objectivity of our knowledge.

  12. 12
    kairosfocus says:

    ES,

    good points, we are by our nature subjects and so have to deal with our finite, fallible, struggling, too often ill-willed selves. That’s why we need that accountable, reasonably reliable warrant for our opinions that moves them to the domain of knowledge.

    Such warrant is in many cases tied to the empirical in-common physical world. But also, it involves our inner contemplative world.

    This is why first principles of right reason, self evident truths and well established facts are so important for warrant — lest we impose crooked yardsticks as standards of straightness, uprightness and accuracy; which would lock out what is genuinely such.

    Hence, the significance of plumbline, self evident truths that are NATURALLY straight and upright. Such then obtains with redoubled force when we deal with abstracta such as the phenomena of Mathematics.

    And so we have the paradox that we are subjects who, to respond to the moral government of duty to truth and to reason, must strain towards the objective.

    And, ironically, a first self-evident, plumbline truth is Josiah Royce’s point that error exists. This turns out to be undeniably true as to say E then assert ~E implies that one of these must fail to refer to reality accurately. On inspection, instantly it is ~E: it is an error to assert that error exists defeats itself. So, too, any scheme of thought that denies objective or even self-evident truth and thus also warrant to undeniable or even just reliable certainty, is defeated, is false.

    This sweeps away vast swathes of currently popular thought and not a little of academic discourse.

    KF

  13. 13
    kairosfocus says:

    F/N: Phil Wilson here, has something interesting, too:

    I want to ask “what does the applicability of mathematics have to say about the foundations of mathematics?” In asking this question I take for granted that there is no serious disagreement about whether mathematics is applicable: the entire edifice of modern science and technology, depending heavily as it does on the mathematisation of nature, bears witness to this fact.

    So what can a formalist say to explain the applicability of mathematics? If mathematics really is nothing other than the shuffling of mathematical symbols in the world’s longest running and most multiplayer game, then why should it describe the world? What privileges the game of maths to describe the world rather than any other game? Remember, the formalist must answer from within the formalist worldview, so no Plato-like appeals to a deeper meaning of maths or hidden connection to the physical world is allowed. For similar reasons, the logicists are left floundering, for if they say “well, perhaps the universe is an embodiment of logic”, then they are tacitly assuming the existence of a Platonic realm of logic which can be embodied. This turns logicism into a mere branch of platonism, which, as we will shall see below, comes with its own grave problems. Thus for both formalists and non-platonist logicists the very existence of applicable mathematics poses a problem apparently fatal to their position.

    Math facts, empirical and abstract alike clearly challenge those who would isolate Math behind an ugly gulch. He keeps going:

    Neither logicism nor formalism is widely believed any more, despite the cliché that mathematicians are platonists during the week and formalists at the weekend. Both perspectives fell out of favour for reasons other than the potentially fatal one about the applicability of mathematics, reasons largely connected with the work of Gödel, Thoralf Skolem, and others. (See Gödel and the limits of logic.)

    Constructivism

    Is the world inherently mathematical or is maths a construct of the human mind?

    The third proposed foundation, intuitionism, never really garnered much support in the first place. To this day, it is muttered about in dark tones by most working mathematicians, if it is considered at all. What is seen as a highly restricted toolkit for proofs and a bizarre notion of limbo, in which a statement is neither true nor false until a proof has been constructed one way or the other, make this viewpoint unattractive to many mathematicians.

    However, the central idea of the enumerable nature of processes in the universe appears to be deduced from reality. The physical world, at least as we humans perceive it, seems to consist of countable things and any infinity we might encounter is a result of extending a counting process. In this way, perhaps intuitionism is derived from reality, from the apparently at-most-countably infinite physical world. It appears that intuitionism offers a neat answer to the question of the applicability of mathematics: it is applicable because it is derived from the world.

    However, this answer may fall apart on closer inspection. For a start, there is much in modern mathematical physics, including for example quantum theory, which requires notions of infinity beyond the enumerable. These aspect may therefore lie forever beyond the explicatory power of intuitionistic mathematics.

    The countable transfinite only touches on the doorstep of that domain. Start with the continuum. He goes on:

    There is one modern idea which could benefit from the finitist logic of the intuitionists: so-called digital physics. It holds that the Universe is akin to a giant computer. The fundamental particles, for example, are described by the quantum state they happen to be in at a given moment, just as the bit from computer science is defined by its value of 0 or 1. Just like a computer, the Universe is based on information about states and its evolution could in theory be simulated by a giant computer. Hence the digital physics motto, “It from bit”.

    But this world view too fails to be truly intuitionistic and seems to sneak in some platonic ideas. The bit of information theory seemingly posits a platonic existence of information from which the physical world is derived.

    But more fundamentally, intuitionism has no answer to the question of why non-intuitionistic mathematics is applicable. It may well be that a non-intuitionistic mathematical theorem is only applicable to the natural world when an intuitionistic proof of the same theorem also exists, but this has not been established. Moreover, although intuitionistic maths may seem as if it is derived from the real world, it is not clear that the objects of the human mind need faithfully represent the objects of the physical Universe. Mental representations have been selected for over evolutionary time, not for their fidelity, but for the advantage they gave our forebears in their struggles to survive and to mate.

    We see here Haldane’s challenge surfacing.

    Wilson continues:

    Created in the image of mathematics

    Formalism and logicism have failed to answer our big question. The jury is out on whether inuitionism might do so, but huge conceptual difficulties remain. What, then, of Platonism?

    Platonists believe that the physical world is an imperfect shadow of a realm of mathematical objects (and possibly of notions like truth and beauty as well). The physical world emerges, somehow, from this platonic realm, is rooted in it, and therefore objects and relationships between objects in the world shadow those in the platonic realm. The fact that the world is described by mathematics then ceases to be a mystery as it has become an axiom: the world is rooted in a mathematical realm.

    But even greater problems then arise: why should the physical realm emerge from and be rooted in the platonic realm? Why should the mental realm emerge from the physical? Why should the mental realm have any direct connection with the platonic? And in what way do any of these questions differ from those surrounding ancient myths of the emergence of the world from the slain bodies of gods or titans, the Buddha-nature of all natural objects, or the Abrahamic notion that we are “created in the image of God”?

    Indeed, the belief that we live in a divine Universe and partake in a study of the divine mind by studying mathematics and science has arguably been the longest-running motivation for rational thought, from Pythagoras, through Newton, to many scientists today. “God”, in this sense, seems to be neither an object in the space-time world, nor the sum total of objects in that physical world, nor yet an element in the platonic world. Rather, god is something closer to the entirety of the platonic realm. In this way, many of the difficulties outlined above which a platonist faces are identical with those faced by theologians of the Judeo-Christian world — and possibly of other religious or quasi-religious systems.

    The secular icon Galileo believed that the “book of the universe” was written in the “language” of mathematics — a platonic statement begging an answer (if not the question) if ever there was one. Even non-religious mathematical scientists today regularly report feelings of awe and wonder at their explorations of what feels like a platonic realm — they don’t invent their mathematics, they discover it. Paul Davies goes further in The Mind of God, and highlights the two-way nature of this motivation. Not only may a mathematician be driven to understand mathematics in a bid to glimpse the mind of God (a non-personal God like that of Spinoza or Einstein), but our very ability to access this “key to the universe” suggests some purpose or meaning to our existence.

    In fact, the hypothesis that the mathematical structure and physical nature of the universe and our mental access to study both is somehow a part of the mind, being, and body of a “god” is a considerably tidier answer to the questions of the foundation of mathematics and its applicability than those described above. Such a hypothesis, though rarely called such, has been found in a wide variety of religious, cultural, and scientific systems of the past several millenia. It is not natural, however, for a philosopher or scientist to wholeheartedly embrace such a view (even if they may wish to) since it tends to encourage the preservation of mystery rather than the drawing back of the obscuring veil.

    Food for thought.

    KF

  14. 14
    kairosfocus says:

    PS: This comment in PW’s thread is very interesting, not least because it bridges to ID issues quite directly:

    In defense of the evolutionary approach
    Permalink Submitted by Anonymous on July 1, 2011

    Thank you for answer Phil. If you don’t mind, I would like to defend my evolutionary explanation a bit (neglecting the other parts of your answer for the moment).

    In a way, there is lot of applied mathematics in biology – a spectacular example would be echolocation in bats or dolphins but there are many others. Admittedly, those are mostly specialized systems and not all-purpose processors like the human mind, but I still think they illustrate that advanced mathematical abilities can be acquired by evolution.

    The human ability to apply mathematical rules and – probably more important for real mathematics – to combine them freely might be a chance product of their coexistence in the same mental apparatus. The chance element is of course a weak point here, but then again, chance is an essential part of evolution and there are quite a few examples where unsuspecting inventions have been coopted by evolution for some unforeseen purpose (e.g. feathers for flight).

    I would also like to outline a possible explanation for the ability of mathematics to work in counter-intuitive realms which is in line with an evolutionary approach: The rules used to interpret the physical world (or to manipulate it) would be more useful the more general (and accurate) they were. I would further posit that intuition is based on what we can directly experience through our senses or actions but those experiences are constrained by the implementation of our physical body. However, mental rules that are able to interpret data accurately in the working range of our body do not necessarily fail where the body fails, in the same way that the validity of a temperature scale does not stop where a particular thermometer reaches the end of its dynamic range.

    I will let this one float for a moment, noting that this illustrates the mindset we face — and that the implicit incoherence of trying to account for rational contemplation on blind computational substrates allegedly organised and programmed by lucky noise that worked is simply missed.

  15. 15
    kairosfocus says:

    F/N: I have added from Nesher on Godel’s epistemology, an illustration of three views. KF

  16. 16
    Eugene S says:

    KF

    Yes, from the point of view of reasonably reliable warrant concerning natural phenomena of this world, I agree with you. However, there is reality that cannot be covered by any reasonably reliable warrant whatsoever. Uncreated divine energy of God that has brought the world into being and sustains it, is not subject to analysis because it surpasses any human reason, as St Dionysius the Ariopagite points out.

  17. 17
    kairosfocus says:

    ES, is that the Dionysius of Ac 17? I’d love to see writings tracing to him. God is of course the ultimate and ultimately rational mind. We can catch a glimpse of a shadow, at least and in that sense think his thoughts after him, being in his image. KF

  18. 18
    Eugene S says:

    KF,

    Furthermore, while man can experience Divine grace, the nature of God is completely unknowable.

  19. 19
    Eugene S says:

    KF,

    Yes, it is him. Even though there is on-going debate among contemporary historians about the attribution of the Corpus to him, Tradition points to him. I go with the Tradition 🙂

    We do not know the nature of God, nor can we in principle. It is completely beyond man.

  20. 20
    kairosfocus says:

    ES, while this is not a theology thread, I think you may be alluding to say Isa 55, where God’s ways and thoughts are as high beyond ours as the heavens are above the earth; also 1 Cor 1 which speaks to the frustration of human wisdom in inquiring on the ultimate truth of God in the context of the gospel. That said, rational communication between God and us is possible per the principle of revelation and that of prayer; the very texts in question claiming to be just such. So, I think some aspects of the Divine nature are cognisable for us, as say Rom 1:18 – 20 and 28 ff suggest also; implying we know enough to lack excuse for our wrongful behaviour and rebellion. But we cannot know him from our own resources as he fully is, we will be baffled by the gap. Back to Math. KF

  21. 21
    kairosfocus says:

    Can I get a link?

  22. 22
  23. 23
    PaoloV says:

    Very interesting exchange of insightful comments between bornagain77, kairosfocus and Eugene S.
    Well done! Thanks.

  24. 24
    kairosfocus says:

    F/N: Time to pick up that comment, on points:

    >>In a way, there is lot of applied mathematics in biology – a spectacular example would be echolocation in bats or dolphins but there are many others.>>

    1 –> Left off, for some bats and cetateans, apparently, much the same genetic code. Which is a huge problem for convergence that does not include libraries.

    2 –> Let’s chunk it from my bodyplan diversity page in the IOSE:

    a recent peer-reviewed paper discloses that:

    Only microbats and toothed whales have acquired sophisticated echolocation, indispensable for their orientation and foraging. Although the bat and whale biosonars originated independently and differ substantially in many aspects, we here report the surprising finding that the bottlenose dolphin, a toothed whale, is clustered with microbats in the gene tree constructed using protein sequences encoded by the hearing gene Prestin.

    (Ying Li, Zhen Liu, Peng Shi, and Jianzhi Zhang, “The hearing gene Prestin unites echolocating bats and whales,” Current Biology, Vol. 20(2):R55-R56 (January, 2010) (internal citations removed. HT: ENV).)

    The restrained language of the journal article softens the force of the point. A Science Daily report brings out the significance more directly:

    two new studies in the January 26th issue of Current Biology, a Cell Press publication, show that bats’ and whales’ remarkable [[sonar echolocation] ability and the high-frequency hearing it depends on are shared at a much deeper level than anyone would have anticipated — all the way down to the molecular level.

    “The natural world is full of examples of species that have evolved similar characteristics independently, such as the tusks of elephants and walruses,” said Stephen Rossiter of the University of London, an author on one of the studies. “However, it is generally assumed that most of these so-called convergent traits have arisen by different genes or different mutations. Our study shows that a complex trait — echolocation — has in fact evolved by identical genetic changes in bats and dolphins.”

    A hearing gene known as prestin in both bats and dolphins (a toothed whale) has picked up many of the same mutations over time, the studies show. As a result, if you draw a phylogenetic tree of bats, whales, and a few other mammals based on similarities in the prestin sequence alone, the echolocating bats and whales come out together rather than with their rightful evolutionary cousins.

    Both research teams also have evidence showing that those changes to prestin were selected for, suggesting that they must be critical for the animals’ echolocation for reasons the researchers don’t yet fully understand.

    Or, more directly, yet, we can plainly see the issue of common design for a common function, across highly specialised animals in widely divergent taxonomic groups. For the likelihood of “identical genetic changes” to a given gene in such divergent groups, to produce sonar systems in water-living and flying mammals being by accident is rather small. So plainly small in fact that a Current Biology journal review suggested that:

    Whales and dolphins belong to the order Cetartiodactyla, and their closest living relatives may be hippopotamuses. Nevertheless, dolphins and porpoises share at least 14 derived amino acid sites in prestin with echolocating bats, including 10 shared with the highly specialised CF bats. Consequently, dolphins and porpoises form a sister group to CF bats in a phylogenetic analysis of prestin sequences (Figure 1). This finding is arguably one of the best examples of convergent molecular evolution discovered to date, and is exceptional because it is likely to be adaptive, driven by positive selection.

    (Gareth Jones, “Molecular Evolution: Gene Convergence in Echolocating Mammals,” Current Biology, Vol. 20(2):R62-R64 (January, 2010) (internal citations removed). HT: ENV.)

    3 –> That compounds the FSCO/I challenge of getting there.

    4 –> to do ecolocation, these animals have to emit signals and process echoes to create a sound-space world model, implying a huge amount of processing.

    >> Admittedly, those are mostly specialized systems and not all-purpose processors like the human mind, but I still think they illustrate that advanced mathematical abilities can be acquired by evolution.>>

    5 –> Notice what has been waved away by using a blanket phrase and failing to engage the warrant that per observation such FSCO/I can be generated by blind, needle in haystack search within the time, atomic resources and population genetics involved?

    6 –> Notice, too the slipping in of the conflation of computation with rationally reflective contemplation?

    7 –> this is not to accuse the commenter, it is to highlight how ever so many have been led to think that a blanket myth readily answers all things.

    >>The human ability to apply mathematical rules and – probably more important for real mathematics – to combine them freely might be a chance product of their coexistence in the same mental apparatus.>>

    8 –> Where does responsible, rationally contemplative freedom to do the mathematics come from on the assumptions and mechanisms of evolutionary materialism and fellow travellers? Ninety-odd years after Haldane wrote, ordinary people should be at least familiar with the seriousness of the issue:

    “It seems to me immensely unlikely that mind is a mere by-product of matter. For if my mental processes are determined wholly by the motions of atoms in my brain I have no reason to suppose that my beliefs are true. They may be sound chemically, but that does not make them sound logically. And hence I have no reason for supposing my brain to be composed of atoms. In order to escape from this necessity of sawing away the branch on which I am sitting, so to speak, I am compelled to believe that mind is not wholly conditioned by matter.” [“When I am dead,” in Possible Worlds: And Other Essays [1927], Chatto and Windus: London, 1932, reprint, p.209. (NB: DI Fellow, Nancy Pearcey brings this right up to date (HT: ENV) in a current book, Finding Truth.)]

    >>The chance element is of course a weak point here,>>

    9 –> Understatement of the year.

    >> but then again, chance is an essential part of evolution>>

    10 –> A huge explanatory gap once one understands why FSCO/I naturally comes in deeply isolated islands in huge configuration spaces that start at 3.27 * 10^150 cells, for just 500 bits. Multiple, correctly arranged, oriented and coupled parts just to achieve function.

    >> and there are quite a few examples where unsuspecting inventions have been coopted by evolution for some unforeseen purpose (e.g. feathers for flight).>>

    11 –> Blissfully unaware of just how outrageously just so story this “standard” answer to irreducible complexity is. He needs to be aware of Menuge’s five challenges:

    IC is a barrier to the usual suggested counter-argument, co-option or exaptation based on a conveniently available cluster of existing or duplicated parts. For instance, Angus Menuge has noted that:

    For a working [bacterial] flagellum to be built by exaptation, the five following conditions would all have to be met:

    C1: Availability. Among the parts available for recruitment to form the flagellum, there would need to be ones capable of performing the highly specialized tasks of paddle, rotor, and motor, even though all of these items serve some other function or no function.

    C2: Synchronization. The availability of these parts would have to be synchronized so that at some point, either individually or in combination, they are all available at the same time.

    C3: Localization. The selected parts must all be made available at the same ‘construction site,’ perhaps not simultaneously but certainly at the time they are needed.

    C4: Coordination. The parts must be coordinated in just the right way: even if all of the parts of a flagellum are available at the right time, it is clear that the majority of ways of assembling them will be non-functional or irrelevant.

    C5: Interface compatibility. The parts must be mutually compatible, that is, ‘well-matched’ and capable of properly ‘interacting’: even if a paddle, rotor, and motor are put together in the right order, they also need to interface correctly.

    ( Agents Under Fire: Materialism and the Rationality of Science, pgs. 104-105 (Rowman & Littlefield, 2004). HT: ENV.)

    In short, the co-ordinated and functional organisation of a complex system is itself a factor that needs credible explanation.

    However, as Luskin notes for the iconic flagellum, “Those who purport to explain flagellar evolution almost always only address C1 and ignore C2-C5.” [ENV.]

    >>I would also like to outline a possible explanation for the ability of mathematics to work in counter-intuitive realms which is in line with an evolutionary approach: The rules used to interpret the physical world (or to manipulate it) would be more useful the more general (and accurate) they were.>>

    12 –> Blissfully unaware of what it takes to erect a broad ranging physical theory.

    >> I would further posit that intuition is based on what we can directly experience through our senses or actions but those experiences are constrained by the implementation of our physical body.>>

    13 –> Intuition usually means knowledge that is based on direct insight or awareness. Insofar as we are built to perceive and interact with the world in certain ways, such constraints obtain. But if they undermine responsible, rational freedom — which is simply not grounded — then we are right back at Haldane.

    >> However, mental rules that are able to interpret data accurately in the working range of our body do not necessarily fail where the body fails, in the same way that the validity of a temperature scale does not stop where a particular thermometer reaches the end of its dynamic range.>>

    14 –> Unaware of the GIGO principle and how risky it is to take a program beyond calibrated, validated, reliable range.

    _______________

    Now, yes, it is almost unfair to pick up a random blog comment like this. But we are not attacking an individual, we are seeing how a deeply indoctrinated pattern of thought glides over its gaps and incoherence.

    A lesson.

    KF

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    PaoloV says:

    Kairosfocus:
    Excellent topic, starting article and commentaries.
    Thanks.

  26. 26
    kairosfocus says:

    F/N: Continuing, let me cite from C. John Holcombe on how we in a nutshell came to the Godel-point [link: http://www.textetc.com/theory/.....atics.html ]:

    Though mathematics might seem the clearest and most certain kind of knowledge we possess, there are problems just as serious as those in any other branch of philosophy. What is the nature of mathematics? In what sense do its propositions have meaning? . . . .

    Even Russell saw the difficulty with set theory. We can distinguish sets that belong to themselves from sets that do not. But what happens when we consider the set of all sets that do not belong to themselves? Mathematics had been shaken to its core in the nineteenth century by the realization that the infallible mathematical intuition that underlay geometry was not infallible at all. There were space-filling curves. There were continuous curves that could be nowhere differentiated. There were geometries other than Euclid’s that gave perfectly intelligible results. Now there was the logical paradox of a set both belonging and not belonging to itself. Ad-hoc solutions could be found, but something more substantial was wanted. David Hilbert (1862-1943) and his school tried to reach the same ends as Russell, but abandoned some of the larger claims of mathematics. Mathematics was simply the manipulation of symbols according to specified rules. The focus of interest was the entities themselves and the rules governing their manipulation, not the references they might or might not have to logic or to the physical world.

    In fact Hilbert was not giving up Cantor’s world of transfinite mathematics, but accommodating it to a mathematics concerned with concrete objects. Just as Kant had employed reason to categories beyond sense perceptions — moral freedoms and religious faith — so Hilbert applied the real notions of finite mathematics to the ideal notions of transfinite mathematics.

    And the programme fared very well at first. It employed finite methods — i.e. concepts which could be insubstantiated in perception, statements in which the statements are correctly applied, and inferences from these statements to other statements. Most clearly this was seen in classical arithmetic. Transfinite mathematics, which is used in projective geometry and algebra, for example, gives rise to contradictions, which makes it all the more important to see arithmetic as fundamental. But of course non-elementary arithmetic is not straightforward, and a formalism had to be developed. H.B. Curry was stricter and clearer than Hilbert is this regard, and used (a) terms {tokens (lists of objects), operations (modes of combination) and rules of formation} (b) elementary propositions (lists of predicates and arguments), and (c) elementary theorems {axioms (propositions true unconditionally) and rules of procedure}. But Volume I of Hilbert and Bernays’s classic work had been published, and II was being prepared when, in 1931, Gödel’s second incompleteness theorem brought the programme to an end. Gödel showed, fairly simply and quite conclusively, that such formalisms could not formalize arithmetic completely.

    What does this mean? Suppose we postulate an arithmetical expression called X. Traditional mathematics would prove X to be either true or false. If different mathematical routes taken within the system proved that X was both true and false, however, then the system was inconsistent. If X could neither be proved as true or false within the system — and the emphasis is crucial, as the consistency could be proved in other ways — then the system is incomplete. Gödel showed that there would always be propositions that were true, but which could not be deduced from the axioms

    In a sense, this hit almost as hard as finding out that Euclidean is not the only possible, consistent, relevant geometry.

    Other schools of thought:

    For intuitionists like L.E.J. Brouwer (1881-1966) the subject matter of mathematics is intuited non-perceptual objects and constructions, these being introspectively self-evident. Indeed, mathematics begins with a languageless activity of the mind which moves on from one thing to another but keeps a memory of the first as the empty form of a common substratum of all such moves. Subsequently, such constructions have to be communicated so that they can be repeated — i.e. clearly, succinctly and honestly, as there is always the danger of mathematical language outrunning its content.

    How does this work in practice? Intuitionist mathematics employs a special notation, and makes more restricted use of the law of the excluded middle (that something cannot be p’ and not-p’ at the same time). A postulate, for example, that the irrational number pi has an infinite number of unbroken sequences of a hundred zeros in its full expression would be conjectured as undecidable rather than true or false. But the logic is very different, particularly with regard to negation, the logic being a formulation of the principles employed in the specific mathematical construction rather than applied generally . . . .

    Social constructivists took a very different line. {13} Mathematics is simply what mathematicians do. Mathematics arises out of its practice, and must ultimately be a free creation of the human mind, not an exercise in logic or a discovery of preexisting fundamentals. True, mathematics does tell us something about the physical world, but it is a physical world sensed and understood by human beings, as Kant pointed out long ago. Perhaps, somewhere in the universe, evolution has made very different creatures, when their mathematics will not resemble ours at all: it is surely possible . . . .

    Jungian psychiatrists regarded numbers as archetypes, autonomous and self-organizing entities buried deep in the collective unconscious. Scientists and mathematicians have found that approach much too shadowy, lacking real evidence or explanatory power. But numbers as predispositions of inner body processes have reappeared in metaphor theory, this time supported by clinical study. Lakoff and Núñez analyze the mathematical metaphors behind arithmetic, symbolic logic, sets, transfinite numbers, infinitesimals, and calculus, ending with Euler’s equation, where e, i and pi are shown to be arithmetizations of important concepts: recurrence, rotation, change and self-regulation. Mathematics is thus a human conceptualization operating with and limited to the brain’s cognitive mechanisms. We cannot know if other (non-human) forms of mathematics exist, and mathematics is the language of science because both disciplines are mappings of source observations onto target abstractions, i.e. brain operations that employ innate and learned understandings of the world around us . . .

    This allows us to see where people have come out along major lines of thought.

    I would suggest, that a proposition is undecidable on a given axiomatic system does not entail that in itself it does or does not capture an accurate description of some feature of structure and quantity. Similarly, while intelligent lobster-armed gorillas from Orion’s Belt may well use different symbols and ways to express themselves, so long as they deal with the logic of being and then with abstract model worlds expressing structure and quantity generally tied to numbers, spatial entities and relationships, there will be much in common.

    At minimum,just to illustrate:

    || + ||| –> |||||

    will still work.

    And in that lies a key point on the sense of the claim that mathematics is based on objective truth.

    KF

  27. 27
    Wouter van Niekerk says:

    KF “lobster-armed gorillas” is that a geographical trait?

  28. 28
    kairosfocus says:

    WvN, A long time ago I read a Sci Fi comic where the alien met that description; I put in Orion’s Belt as that is a fairly nearby star-forming region in that famed constellation on the celestial equator . . . cosmology not geography. I am just suggesting an alien with a very different take on things — core of math facts would be much the same. KF

  29. 29
    kairosfocus says:

    F/N: Let’s see how IEP describes Mathematical Platonism (where, no, this is not equal to Plato’s theory of forms):

    Traditionally, mathematical platonism has referred to a collection of metaphysical accounts of mathematics, where a metaphysical account of mathematics is one that entails theses concerning the existence and fundamental nature of mathematical ontology. In particular, such an account of mathematics is a variety of (mathematical) platonism if and only if it entails some version of the following three Theses:

    1] Existence: Some mathematical ontology exists.
    2] Abstractness: Mathematical ontology is abstract.
    3] Independence: Mathematical ontology is independent of all rational activities, that is, the activities of all rational beings . . . .

    platonists have maintained that the items that are fundamental to mathematical ontology are objects, where an object is, roughly, any item that may fall within the range of the first-order bound variables of an appropriately formalized theory and for which identity conditions can be provided . . . object platonism is the conjunction of three theses: some mathematical objects exist, those mathematical objects are abstract, and those mathematical objects are independent of all rational activities. In the last hundred years or so, object platonisms have been defended by Gottlob Frege [1884, 1893, 1903], Crispin Wright and Bob Hale [Wright 1983], [Hale and Wright 2001], and Neil Tennant [1987, 1997].

    Nearly all object platonists recognize that most mathematical objects naturally belong to collections (for example, the real numbers, the sets, the cyclical group of order 20). To borrow terminology from model theory, most mathematical objects are elements of mathematical domains. Consult Model-Theoretic Conceptions of Logical Consequence for details. It is well recognized that the objects in mathematical domains have certain properties and stand in certain relations to one another. These distinctively mathematical properties and relations are also acknowledged by object platonists to be items of mathematical ontology.

    More recently, it has become popular to maintain that the items that are fundamental to mathematical ontology are structures rather than objects. Stewart Shapiro [1997, pp. 73-4], a prominent defender of this thesis, offers the following definition of a structure:

    I define a system to be a collection of objects with certain relations. … A structure is the abstract form of a system, highlighting the interrelationships among the objects, and ignoring any features of them that do not affect how they relate to other objects in the system.

    According to structuralists, mathematics’ subject matter is mathematical structures. Individual mathematical entities (for example, the complex number 1 + 2i) are positions or places in such structures . . .

    These ideas of course arise from Mathematical practice, where we find ourselves dealing with abstracta and find ourselves constrained by facts tied to them and by logical relationships.

    This sort of thing does not sit easily with a day and age incluned to evolutionary materialism and empiricism, while being haunted by Kant’s ugly gulch. A good example is how often we find ourselves tempted to reduce minded, responsible, rational contemplation to computation. Never mind the ugly gulch between blind GIGO limited mechanical and stochastic processes and the rational, responsible freedom required for mind to be coherent, and consequences of self-referentiality.

    Another issue is, what is truth vs what is it that we have warranted as true. So, is there a string of ten or a hundred or a thousand zero’s or 1’s etc in the expansion of pi? That we may never know is different from there is or is not in some abstract sense. But many are inclined to think that until we have constructed a solution that manifests such a string, it is neither there nor not there.

    That is, we here see a challenge to the law of the excluded middle, a key logical principle connected to distinct identity. Such then feeds into the view we have seen, where Mathematics is whatever it is Mathematicians as a circle of subjects do and accept.

    I tend to see that we may err and have rather bounded rationality, but that does not affect what is or may be beyond the circle of what we know or may ever know. But of course, that cuts across the empiricist spirit of the age.

    On the other hand, that such are forced to traffic in mathematical abstracta to practice science may well be a corrective message.

    The plumb-line speaking to the crooked yardstick, in short.

    So, I find myself drawn to Godel’s option C as was added to the OP; noting that there is a body of specific math facts and linked observable phenomena that beg for structuring through axiomatisation and systematic derivation of theorems and results, but also once a scheme is consistent such systems cannot be complete.

    I also draw broad confidence in grand coherence from the way vast domains are locked together to infinite precision by our old friend, 0 = 1 + e^i*pi.

    Where also, once a wider world of reality is, all that is must be consistent with all else that is. And any one thing is consistent within its own existence and core character. That is, there can be circles and squares but no circle squares.
    KF

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