Over recent days, there has been an exchange at UD on the objectivity vs subjectivity of mathematical knowledge. This is relevant to our understanding of knowledge, and to our recognition of the credibility of Mathematical findings on debated matters. This instantly means that the specific concern and the penumbra of generalised perceptions of Mathematics, Science and objectivity of knowledge are relevant to the ID debate. So, it is appropriate to clip from the discussion in the axioms of math thread.

First, BO’H and his suggestion that he and I actually in the end agree:

BO’H, 34: >>[to:] EricMH – I believe that mathematics, in different respects, is both subjective and objective.

I responded in 35, but I think it is appropriate to note, on the framework/ structure of mathematical knowledge:

KF, 24: >>The first locus of warranted, accountable truth in a novel domain of structure and quantity (i.e. Mathematics, thus defined insofar as succinct definition is possible) will be the logical import of the framework so laid out in axioms, elaborated in proofs of key theorems and derivation of onward results in accord with logical necessity:

If [AXIOMS] Ax1, Ax2 . . . Axn then perforce

[THEOREMS:] Th1, Th2 . . . Thm.

Per Axi and Thj, Th k etc then also

[RESULTS OF INTEREST/ APPLICATIONS:] Res1, Res2 . . . Resq (which may be of practical interest)

and so forth.

This already imports a huge body of controlling, structuring, accountable facts, principles, patterns and results from the long historical and logical core of mathematics (and logic).

All of this, in a context where numbers, structures, frameworks from across the domains and the raw reality of what is logically possible constrain the work to conform to the states of affairs in physically actualised and abstract worlds defined by things such as key sets [cf. above on numbers great and small], algebraic structures [classically, groups, rings, fields . . . which have such technical meanings that one has to be careful not to invoke them unintentionally in the discussion] and entire sub-disciplines of Mathematics.

If such were not so, we would not have a discipline, we would have a chaos of incoherent, incompatible, unreliable claims, counter-claims and much more.>>

Where, too, I note on the meaning of objective vs subjective:

KF, PPS to OP: >>Collins English Dictionary is helpful:

subjective

adj

1. belonging to, proceeding from, or relating to the mind of the thinking subject andnotthe nature of the object being considered

2. of, relating to, or emanating from a person’s emotions, prejudices, etc: subjective views.

objective

adj

1. (Philosophy) existing independently of perception or an individual’s conceptions: are there objective moral values?.

2. undistorted by emotion or personal bias>>

Mathematicians are subjects, but the discipline and its body of knowledge will be objective through having accountable, logically driven warrant and the need to connect even novel axiomatic systems to the core body of facts on structure and quantity, starting with numbers.

It will be helpful to throw a side-light from Wikipedia (speaking against ideological tendency), on Groups:

KF, 13: >>To give an idea, let me select a typical algebraic structure:

In mathematics,

a groupis an algebraic structure consisting of a set of elements equipped with an operation that combines any two elements to form a third element and that satisfies four conditions called the group axioms, namely closure, associativity, identity and invertibility.One of the most familiar examples of a group is the set of integers together with the addition operation, but the abstract formalization of the group axioms, detached as it is from the concrete nature of any particular group and its operation, applies much more widely[–> there is a back-trail and pattern of structure right there, and it goes back beyond axiomatisation to an ancient pool of knowledge of mathematical facts etc]. It allows entities with highly diverse mathematical origins in abstract algebra and beyond to be handled in a flexible way while retaining their essential structural aspects. The ubiquity of groups in numerous areas within and outside mathematics makes thema central organizing principle of contemporary mathematics.[1][2] [–> notice the role of coherence, the group ties a lot of things together]Groups share a fundamental kinship with the notion of symmetry. For example, a symmetry group encodes symmetry features of a geometrical object: the group consists of the set of transformations that leave the object unchanged and the operation of combining two such transformations by performing one after the other. Lie groups are the symmetry groups used in the Standard Model of particle physics; Poincaré groups, which are also Lie groups, can express the physical symmetry underlying special relativity; and point groups are used to help understand symmetry phenomena in molecular chemistry.

The concept of a group arose from the study of polynomial equations, starting with Évariste Galois in the 1830s. [–> again, reaching back to roots] After contributions from other fields such as number theory and geometry,the group notion was generalized and firmly established around 1870[–> notice, the pattern came first and led the structuring definitions] . Modern group theory—an active mathematical discipline—studies groups in their own right.>>15: >>[To illustrate:] For integers, closure: a + b is an integer too, say c. There is an identity element 0 such that a + e = a. Likewise associativity: (a + b) + c = d = a + (b + c). For each a there is a b so that invertibility: a + b = 0, i.e. b is -a. This pattern can be generalised, and it opens up a new world. But that new world will have links tracing back into the logical and often the historic core of mathematics. Also, there are accountable principles of reasoning that allow reliable argument and accurate conclusions that are in principle accessible to anyone.>>

With such background in hand, we may now consider:

KF, 35: >>I begin to think we are thinking of different things; and that the story of the blind men and the elephant or the metaphor of climbers attacking a new mountain may speak.

Explorers are subjects, knowers are subjects (one has to believe to know, it is the warrant that lends credibility). Likewise, the choice of angle of attack may reflect individual, institutional or cultural process, though the pattern that many separate workers pick up a fresh area in a given time [Quaternions are a case in point, and the handing on to the somewhat derivative vector approach] also suggests that opportunity reflects presence of a common reality.

On the other hand, the elephant and the mountain — here, the substantial domain of quantity and structure, the effect of logic and the effect of connectivity back to the heart of mathematical reality — are very much out there and have properties of their own.

Then, there is the impact of Godel’s incompleteness, which shows that there are truths in an axiomatic domain that are not reachable from a given set of axioms. This directly implies that there are structures, quantities and properties that are independent of our conceptualisation and exploration.

So, when axiomatisation sets out a new field of exploration, it is not like writing a new work of say Sci Fi literature that is then studied as a creation of an author. Different authors create different “universes” and even if other authors join in, some effort has to be made to keep things coherent and realistic. Others will naturally create very different universes, as is a very familiar pattern from the many series of novels that are out there. A domain that is at heart creative will naturally be divergent.

By contrast, explorers of a new mountain or territory find that they converge on a consistent map. The blind men come to a consistent pattern about the elephant as they move about the beast and share then unify perspectives. (The story deliberately exaggerates the differences due to partial exploration.)

Now, too, we live in a day and age in which subjectivism and relativism are dominant. There is a natural tendency to exaggerate the significance of subjectivity of persons (where, ironically evolutionary materialism actually undermines the responsible rational freedom that lies at the heart of being a subject). There is a tendency to reduce just about everything to a narrative of power and marginalisation or oppression, with a myth of fundamentally socialistic, statist liberation. (Ironically, the powerful state is perhaps the single greatest threat to freedom.) And lurking deep underneath is the Kantian ugly gulch beteweeen the inner phenomenal world and the outer, extra-mental world of things in themselves. Where, it is commonly perceived that we project a creative rationalisation to the outer world, but cannot actually come to know it as it is in any material degree.

The last is perhaps the most subtle influence, but its worldview shaping power is enormous. That is why I have taken time to repeatedly share F H Bradley’s opening remarks from his Appearance and Reality. The very name of the book speaks.

The essential point is that there is a fatal self-referentiality in making a Kantian ugly gulch claim. To claim to know the un-knowability of external reality as it actually is (as opposed to how it appears) is to claim to know just such an external reality. Namely, the alleged un-knowability characteristic property. So, we have self-referential incoherence and self-falsification. And indeed, we also see a case where, never mind whatever distortions our organs of sensing and perceiving may impose, we also have access through logical analysis to key, powerful truth that connects to reality through a contemplative abstract process of insightful reasoning.

A sounder approach, then, is to recognise with Josiah Royce that error exists — and, that this is not merely a perception or a consensus but that it is undeniably, self-evidently true. To try to deny it rapidly turns around and demonstrates the truth of the claim — to suggest it is an error to imagine that errors exist is obviously self-defeating logically (but may happen psychologically or rhetorically).

It is a point of certain knowledge about our interaction with the realities of our world.

Thus, schemes that deny certain knowledge and the possibility of truth as accurately describing reality fall to the ground. Their name is legion.

Instead, we can accept that we have perspectives which are error prone but can also have well warranted access to the credible, reliable truth about much of reality. In some cases to undeniable certainty. The triple first principles of right reason that turn on properties of distinct identity are a crucial further case in point.

As was outlined in the OP that act of recognition of distinct identity immediately leads to two-ness, thence the natural numbers. In 18 above, further points were outlined, to show how we can get to the domain of numbers using several approaches and even complementary definitions, but we also see that numbers all the way to continuum, transfinites, infinitesimals, as well as complex numbers, vectors and similar structures that span abstract spaces in multiple dimensions, are tied to logic and form the substantial heart of Mathematics.

Thus emerges the property of mathematics thatIn exploring it, we may have perspectives and some freedom to set out start points and exploratory projects, but that does not change the underlying common reality that we tend to converge upon and find to have a coherent and consistent unity that is powerful and functional.it is naturally the logic of structure and quantity leading to abstract logically structured model worlds; so that through coherence and common facts and findings Mathematical reality — yes, an abstract reality starting with numbers — has a substantial objectivity.Indeed, Fermat’s last theorem, so-called, was like an irritating bit of sand that sparked worlds of exploration.

Wiki is a useful source for a summary:

In number theory, Fermat’s Last Theorem (sometimes called Fermat’s conjecture, especially in older texts) states that no three positive integers a, b, and c satisfy the equation a^n + b^n = c^n for any integer value of n greater than 2. The cases n = 1 and n = 2 have been known to have infinitely many solutions since antiquity.[1] [–> such is of course an answer to an inviting onward question from Pythagoras’ theorem.]

This theorem was first conjectured by Pierre de Fermat in 1637 in the margin of a copy of Arithmetica where he claimed he had a proof that was too large to fit in the margin. The first successful proof was released in 1994 by Andrew Wiles, and formally published in 1995, after 358 years of effort by mathematicians. The proof was described as a ‘stunning advance’ in the citation for his Abel Prize award in 2016.[2] The proof of Fermat’s Last Theorem also proved much of the modularity theorem and opened up entire new approaches to numerous other problems and mathematically powerful modularity lifting techniques.

The unsolved problem stimulated the. It is among the most notable theorems in the history of mathematics and prior to its proof, it was in the Guinness Book of World Records as the “most difficult mathematical problem”, one of the reasons being that it has the largest number of unsuccessful proofs.[3]developmentof algebraic number theory in the 19th century and the proof of the modularity theorem in the 20th centuryNotice that studied, fence-sitting word, development. Development by inner subjectivity or development by collaborative exploration of a space of abstract but very real possibilities, entities, quantities and structures with substantial properties of their own?

The evidence from the at long last proof, is that the properties inhere in the logic of structure and quantity.

Where, too, there is an implication that the intersection of such abstract links of logical necessity with the physical world leads to the astonishing power of mathematics in the sciences.

Which again underscores that we are grappling with realities that are objective and largely independent of our own subjectivity. Never mind that which projects we pick to explore, how we happen to explore such and the subsequent history and body of credible knowledge will reflect such acts of subjects.

We must not conflate or confuse the two.>>

Food for thought. **END**

**PS:** I think it is also helpful to clip 18 and flesh out with some illustrations, as numbers great and small are the utter core of Mathematics, which I understand to substantially be — compact definition — *the logic of structure and quantity*, which we then create a discipline by studying as best we can. Where it is that substance and that logic of warrant that confers objectivity to the knowledge involved:

KF, 18: >>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.)>>

[to:] kf – yes, some parts of mathematics are objective, I don’t think anyone would disagree with that. But that doesn’t stop subjective choices being made, only that some subjective choices are then (objectively) shown to be a bad idea.

Even if the whole logical structure of mathematics is a coherent whole, the choice of what parts we look at, and how we express that, are (at least in part) subjective. Fermat’s Last Theorem isn’t objectively that important in mathematics, but we know a lot about it because of a subjective feeling that it was important. Thus branches of mathematics that are related to Fermat’s last theorem are relatively well sketched out.>>