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:

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 isbeforeaxiomatisation, 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:]

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:

. Truth says of what is that it is, and of what is not that it is not.)realityis the state of affairs that exists across actualised and abstract worlds,truthaccurately describes some targetted facet of reality to which it refersOn 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.)>>

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

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.

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

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

KF

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

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:

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:

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:

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

A revealing thought-piece: https://www.fordham.edu/info/20603/what_is_mathematics

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

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

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:

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:

>>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

PS: An instructive syllabus is here: http://www.uio.no/studier/emne.....2;copy.pdf

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.

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

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

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

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

We see here Haldane’s challenge surfacing.

Wilson continues:

Food for thought.

KF

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

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.

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

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.

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

KF,

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

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.

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

Can I get a link?

I see: http://www.documentacatholicao.....ks,_EN.pdf

Very interesting exchange of insightful comments between bornagain77, kairosfocus and Eugene S.

Well done! Thanks.

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:

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:

>>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:

>>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

Kairosfocus:

Excellent topic, starting article and commentaries.

Thanks.

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 ]:

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:

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

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

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

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

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

Headlined: https://uncommondescent.com/mathematics/mathematical-realism-platonism-and-nesher-on-godels-option-c/