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.)>>