# BO’H asks: “aren’t the axioms that mathematicians assume subjective? (they may be rational, but they’re not the only possible axioms that could be used)”

This is yet another significant issue that emerges from the ongoing exchanges on subjectivity, objectivity, possibility of objective moral truth, etc. And, the deep interconnectedness of what we are discussing is proving quite fruitful.

So, I think it is useful to now headline Bob’s remark in the rebooting ethics education thread, which ties in Mathematics. And those who find it hard to follow use of indented text blocks to quote, please pardon that praxis:

BO’H, 25 :>>but aren’t the axioms that mathematicians assume subjective? (they may be rational, but they’re not the only possible axioms that could be used) What follows after that is (or at least should be!) objective, of course.>>

My response is:

KF, 26: >>No. Instantly, an axiomatic framework is an element of a modelling exercise that is open to discussion, argument, warrant and required coherence with an existing body of established knowledge.

Coherence of the system is a major constraint on whims and fancies [which is where something would become subjective, captive to a subject’s imagination], and is tied to our rich, deep and widely accessible experience of structure, quantity and linked reasoning. So, for instance, the main structure of mathematics, numbers, emerges from the world-framing fact of distinct identity so that a world

W = {A|~A},

thence two-ness and what flows from that.

Such issues are so deeply embedded that we tend to overlook their significance.

I think I also need to point out that objectivity is not synonymous with either infallibility or absolute truth. Something is infallibly true where it is beyond possibility of error, typically due to some form or other of self-evident undeniability, e.g. that

||| + || –> |||||

which we may symbolise as

3 + 2 = 5.

Absolute truth is the whole, untainted, undiluted correct description of a matter in hand: truth, the whole relevant truth, nothing but the truth. (Hence the famous court room oath.)

By contrast, objective truth is about the degree of truth that is externally accessible (i.e. in principle available to “anyone”) and is so warranted as credible (thus, reliable) that it is is worthy of belief and willingness to act i/l/o the degree of warrant. It is fallible, but a responsible party aware of its warrant should act on it with confidence. (And yes, this all brims over with the inextricably entangled, intertwined hand-in-hand links between the IS and the OUGHT. That is in part why it is so important that we bridge IS and OUGHT at world-root level.)

Knowledge is of course our normal term for warranted credibly true and reliable belief; as “belief” implies it is clearly held by subjects who come to believe. And likewise the warranting process is an active process created by subjects and evaluated by subjects. Subjectivity is inevitable in knowledge but that obviously does not lock out the objective nature of well warranted truth claims. Nor does it reduce knowledge to subjectivism.

Likewise, abstracta are not parts of the world of the direct senses, they are inferred, conceptualised, reflected on. But that does not make them subjective in the relevant sense, i.e. lacking in objective warrant and locked into some individual subject’s perceptions, potential for error, bias, gaps etc. The possibility and actuality of cross checking greatly enhances reliability.

Perhaps a simple exercise similar to Babbage in the ninth Bridgewater Thesis will help.

Suppose subject s1 believes some claim c is true on some point of warrant w that is accessible to other subjects in a relevant community, s1 to sn. Now, w has a probability of error in the case of an individual subject sj, say 0.001. Obviously, providing the odds of error are independent and stable, the cumulative odds of the entire community being in error run like e^n. For n = 3, E_3 = [10^-3]^3 = 10^-9 already. For n = 11, E-11 = 10^-33, and for n = 500, E_500 = 10^-1500. Reliability rises very quickly indeed with multiple independent witnesses with fairly reliable means of warrant. [Of course, with error-prone subjects, the odds of SOMEBODY in the chain/community being wrong also rise with the number. Odds that no-one is in error run like (1 -e)^n, which exponentially falls toward zero.]

Where also, the low likelihood of error in the process itself creates a pattern where a super-majority already will most likely be correct. And of course if the process of warrant is itself accessible to independent cross-checking, odds of overall and consistent error in warrant and/or in conclusion will fall.

Of course, if the community is so interconnected that they reduce to being in effect a single observer, the odds rise back to e. This leads to the significance of multiple, independent lines of warrant w1 to wk that converge on the same verdict: coherence is exponentially mutually reinforcing in squeezing out the likelihood of error. Classically, in the mouth of two or three [independent] witnesses shall a word be established.

Such of course highlights the potential and dangers of a peer-review system, as if the system becomes captive to a common, enforced ideology, its reliability falls drastically due to reducing from multiple, mutually reinforcing lines of convergent warrant to being a single line of claimed warrant.

Nor, does the possibility of multiple options in axioms change the matter of objectivity.

As can be seen with classical Euclidean Geometry, multiple alternative axioms are possible but functionally equivalent (esp. relative to the parallel lines axiom). And where significantly diverse axioms are possible, their effect is to materially change the subject, here, from a Euclidean space to an elliptical or spherical or hyperbolic one, etc. And always, coherence is treasured across domains, so that the system of the logic of structure and quantity is in its heart mutually and freely accessible and reinforcing.

Action of subjects does not automatically force subjectivity of truth claims resulting from such actions, once we have externally accessible warrant and for preference independent, converging lines of mutually reinforcing evidence and argument.

Using the analogy of a spider’s spiral web, the central parts may not be directly accessible to external anchor-points, but they function to tie the whole together through providing a node that unifies the external anchor points, with the spiral web providing a common path from any point to any other.>>

Again, food for thought, especially on the importance and value of diverse schools of thought/paradigms in an intellectual or decision-making community.

Consensus is valuable but there can be such a reality as too much of a good thing. END

PS: A sketch map of the surreals, embracing numbers great and small:

PPS: Collins English Dictionary is helpful:

subjective
1. belonging to, proceeding from, or relating to the mind of the thinking subject and not the nature of the object being considered
2. of, relating to, or emanating from a person’s emotions, prejudices, etc: subjective views.

objective
1. (Philosophy) existing independently of perception or an individual’s conceptions: are there objective moral values?.
2. undistorted by emotion or personal bias

## 35 Replies to “BO’H asks: “aren’t the axioms that mathematicians assume subjective? (they may be rational, but they’re not the only possible axioms that could be used)””

1. 1
kairosfocus says:

Good questions dept:

BO’H asks: “aren’t the axioms that mathematicians assume subjective? (they may be rational, but they’re not the only possible axioms that could be used)”

2. 2
Bob O'H says:

No. Instantly, an axiomatic framework is an element of a modelling exercise that is open to discussion, argument, warrant and required coherence with an existing body of established knowledge.

But how does this mean that the choice of axioms can’t be subjective? We can discuss and argue about lots of things that are subjective.

I’m afraid i don’t see an answer to my question in the rest of your post. My point is that one can chose different axioms and come to different mathematical results (indeed, some mathematical statements are only provable if some specific axioms are chosen). So why, if we are doing mathematics (and thus are not trying to model the real world :-)) do we chose one set of axioms over another?

3. 3
kairosfocus says:

BO’h: the accountability of novel systems through coherence with an existing formal and informal body of knowledge on structure and quantity secures objectivity. Though that is so deeply embedded and often implicit that we may not be aware of it. KF

4. 4
EricMH says:

Subjective implies any set of axioms will. But there is only a very small set of axioms we are interested in, which are the consistent axioms, chosen such that they cannot prove the same thing to be both true and false.

5. 5
Bob O'H says:

kf @ 3 – I’m sorry, but I don’t understand what you mean.

EricMH – indeed. But given there is more than one set of axioms, how do we decide which to investigate?

6. 6
jdk says:

Re 4: yes, axioms that produced contradictory results must be rejected as the basis for a consistent system.

But actually, there are an infinite set of axioms that produce different results. For one example, take the mechanism that produces Mandelbrot’s set, and then start changing the function that is used for the iteration. Given the function as an axiom in the system, there are an infinite number of possible axioms.

Another, simpler example, is Conway’s “Game of Life”. A very simple set of rules applied to a beginning configuration creates a unique result that can only be found by actually stepping through the iterations. Change the rules, and the same beginning configuration produces a different result.

The rules are arbitrary, and some sets of rules produce more interesting results than others, but there are a very large number of possible rules that could be applied to the basic idea.

In neither of these examples is there any possibility of contradictory results.

7. 7
jdk says:

re 5: I also don’t understand 3.

8. 8
groovamos says:

OK so I’m going to put this out because I think it is somewhat relevant to the thread. I’m writing a thesis which I hope to turn into a Ph.D. The topic concerns information theory and measurement and specifically deep into the paper shows a derivation to the statistical parameters of a recieved energy (electromagnetic, acoustic) energy pulse. The derivation is a little mushy but employs a set of convergent characteristics that is a fairly sound approach as I have convinced myself. It is somewhat relevant to the thread because in many places there are no axioms to rely upon that pass down all the way to the end result. I will maybe link to the entire paper at some point but to show what I mean I will link to the introduction which came late in the process. And in which I hit the reader with a bombshell that cannot be found elsewhere in the literature and that is a characterization of Shannon entropy of a sample point of a gaussian process or measure. The reason it cannot be found elsewhere is that it doesn’t seem to be strictly axiomatic by any pathway to derivation. I used the term slightly mushy above and the introduction is quite mushy (heuristic) and relies greatly on analogy as you will see in table 1. I also discuss an elementary connection to epistemic logic in the derivation which should come as no surprise given the requirement of an unavoidable scenario of mind to the application of information theory. The introduction employs no higher mathematics and so should be within scope you can understand. Much of the mushiness comes from an assumption that the Shannon principle of log(1/p) has a more broad application than maybe the man himself realized in the early days, and I quote another source on this in footnote 6. OK so there are a couple of equations that come from further into the paper that are very diffucult to prove, and that is why the page count is currently at 99 pp. But I link to the first 9, where I bring those two equations forward to help with the somewhat mushy lack of rigor and independence on axioms. My email address is at the top where the domain is yahoo.com. I would be interested in any comment here or by email. Here is the link thx guys: https://drive.google.com/open?id=1FQpdxqqKnp5m3oNXRRBuqapmfqLAn0lo

9. 9
kairosfocus says:

BO’H:

Mathematics came first, axiomatisation — even with Geometry — came later. In most cases, C19 – 20. Where, if the proposed axioms were not consistent with what was already known, they would not have succeeded.

And yes, there were some alternatives for systems, such as alternative formulations of the 5th Euclidean Postulate. I have seen an argument to the effect that Pythagoras’ theorem could have stood in for it!

Now, if you move axioms beyond a certain threshold, you change the discipline, as happened with non-Euclidean geometries. However, even so, you depend on logic and on a core of things linked to the number sets [which can be tied back to the triple first principles that hang from distinct identity] and a lot of other things, and indeed, you set up new systems or — better, logical model worlds — in ways that tie back to such core things and the general body of knowledge regarding the logic of structure and quantity.

As just one little example, if you are consistent with core logic, you must also be consistent with the scheme of arithmetic, as the four core operations can be synthesised through logic gates/operations. And, inference rules depend implicitly on those core principles also. In short, unconsciously, we have a lot of testing criteria that have to be met in setting up a new axiom scheme, some formal, some informal and embedded in what it means to have mathematical knowledge.

So, when we set up a novel model logic-world with structures and/or quantitative features of interest [and this specifically includes computer programs and simulation exercises], we are not free to be completely arbitrary, there is accountability and warrant that runs back through to the heart of mathematical knowledge.

And so while there is room for creativity and surprise etc, we are not in the sort of arbitrary, whims and fancies situation that marks the subjective. Objective criteria obtain.

Going to the world of sets that are linked to the Mandelbrot set, of course you set up your computerised model world, and can change relevant functions etc, but again, the connexions back to the core are everywhere, or frankly your program could not be reliable. Let me add, GIGO still rules the roost and brings objectivity to bear, sometimes quite harshly.

Where too, Groov’s convergence of useful but not 100% reliable clues is an example of the rise of a sound picture from enough fairly independent elements.

KF

10. 10
ScuzzaMan says:

@jdk

Have you read Stephen Wolfram’s “A New Kind of Science“?

He might have somewhat to say on the matter of how many rules are possible (he says it’s far from infinite. WRT “Life” he says, iirc, that it’s about 250) and how many might be considered contradictory.

Rather ironically, that whole discussion had something to do with a total of 6 sets of rules that near perfectly replicate the 6 major types of shells.

The rest are open-ended sieves having no utility to a shell-growing mollusc.

The “interesting” part was that all 6 designs (ahem!) appear abruptly and simultaneously and fully-formed in perfect working order in the cambrian (?) and there is no evidence of any of the non-working (uninteresting) designs ever having existed.

Chalk it up to coincidence, eh?

11. 11
kairosfocus says:

BO’H: what do you think of Wolfram’s use a computer to explore a mathematical world approach discussed here? https://www.davidhbailey.com/dhbpapers/dhb-wolfram-2012.pdf KF

12. 12
Amblyrhynchus says:

the accountability of novel systems through coherence with an existing formal and informal body of knowledge on structure and quantity secures objectivity

I don’t see how this get’s us to objectivity. If two mathematicians disagree about including the axiom of choice in the set theory, what objective standard should they refer to? How about when we move from set theory to abstract algbera, we might be able to model the real world with methods from group theory but I don’t think there is any objective basis the the axioms used to define it?

13. 13
kairosfocus says:

Ambly, you will see that I have noted that axiomatic systems create logical model worlds with interesting structural and quantitative features. Crudely, that one can design a sedan car or a pickup truck does not turn the exercise into an at whim in my head game inaccessible to others — not if you want your vehicle to work. Less crudely, that one may create world A or world B as a logical model world does not render these worlds essentially arbitrarily set out imaginative entities that have little or nothing in common cross-worlds. All along, they are constrained by a body of principles, facts, insights and body of knowledge including principles of inference and warrant. For instance, coherence internally and back to the heart of Math and logic are cases in point. It is accessibility and corrigibility of warrant per coherent principles that confers accountability . . . close to the heart of objectivity. When one picks one side or the other of the choice debate, one is setting out one such world or another and they each have domains of interest. KF

PS: To give an idea, let me select a typical algebraic structure:

In mathematics, a group is 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 them a 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.

14. 14
Amblyrhynchus says:

I have to admit I find it hard to extract much from this. But if one was to create a coherent moral code that didn’t refer to any outside source of moral laws wouldn’t that be ‘objective’ by this standard?

15. 15
kairosfocus says:

Ambly, did you see how I spoke about an old friend, Group theory, as a case in point? 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 accessibble to anyone. KF

PS: This thread is not about moral codes, and indeed the earlier discussions were about moral reasoning and key principles, not codes in the sense of a corpus of rules. Rules there may be but they are not essentially arbitrary. of course in morality, a lot of people who do not like a particular result that cuts across what they want to do, will tend to imagine that it is an irrational imposition. You may note that I have discussed the impact of making a crooked yardstick the standard for straightness, accuracy and uprightness which would then make what is actually these things seem to be wrong. This is why we need plumbline test cases that are naturally, patently upright and straight that expose such crookedness.

16. 16
Amblyrhynchus says:

Like I say, I can’t see how any of this gets us to objectivity in the sense that word is usually used. perhaps you can explain it more clearly, but otherwise, I guess I’ll leave you to it.

17. 17
kairosfocus says:

Ambly, objectivity is not a synonym of absolute truth nor of being specifically observed empirical facts, as the OP notes. It instead has to do with warranted claims that are accessible to — accountable before — others, not just perceptions and tastes or personal views of a subject. In Math, axiomatic systems lay out logically driven model worlds that tie back into the overall system of mathematical thought in many ways. Where Mathematics can be understood as the logic of structure and quantity (that emphasis showing the locus of truth, namely that structures of type X and quantities of type Y are interconnected as shown, on force of logical necessity, which will then directly apply to particular circumstances exhibiting such structures and quantities with all the power of that force, e.g. 2 + 3 MUST be 5). Groups being a capital case in point from abstract algebra. The multiple interconnectivity and coherence requisites of Mathematical systems, the links back to the historic and logical core and commonplace facts [such as numbers] and much more — as was seen for groups — confer such objectivity to mathematical claims. Their truth lies in the nature of the [admittedly abstract] entities, not in our particular perceptions, whims and fancies, and we access the warrant of reliable, credible truth through an accountable process of responsible rational inquiry. That’s why axioms cannot be set up willy-nilly without chaos and it is why alternatives may be functionally equivalent or else in effect create a materially diverse model world. E.g., ponder alternate 5th postulates within the Euclidean fold (parallel lines vs angle-sum of a triangle) vs non euclidean Geometries such as elliptical or hyperbolic. And indeed, group patterns are found to apply to significant cases of physical reality. KF

PS: I clip the OP:

Absolute truth is the whole, untainted, undiluted correct description of a matter in hand: truth, the whole relevant truth, nothing but the truth. (Hence the famous court room oath.)

By contrast, objective truth is about the degree of truth that is externally accessible (i.e. in principle available to “anyone”) and is so warranted as credible (thus, reliable) that it is is worthy of belief and willingness to act i/l/o the degree of warrant. It is fallible, but a responsible party aware of its warrant should act on it with confidence. (And yes, this all brims over with the inextricably entangled, intertwined hand-in-hand links between the IS and the OUGHT. That is in part why it is so important that we bridge IS and OUGHT at world-root level.)

Knowledge is of course our normal term for warranted credibly true and reliable belief; as “belief” implies it is clearly held by subjects who come to believe. And likewise the warranting process is an active process created by subjects and evaluated by subjects. Subjectivity is inevitable in knowledge but that obviously does not lock out the objective nature of well warranted truth claims. Nor does it reduce knowledge to subjectivism.

Likewise, abstracta are not parts of the world of the direct senses, they are inferred, conceptualised, reflected on. But that does not make them subjective in the relevant sense, i.e. lacking in objective warrant and locked into some individual subject’s perceptions, potential for error, bias, gaps etc. The possibility and actuality of cross checking greatly enhances reliability.

18. 18
kairosfocus says:

PPS: 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.)

19. 19
kairosfocus says:

F/N: I have added to the OP a PPS that cites Collins English Dictionary on a particularly helpful understanding of subjective vs objective. It seems basic conception of core ideas is a key factor in all of the above. KF

20. 20
LocalMinimum says:

KF:

Enjoying the posts.

Amblyrhynchus @ 12:

I’ll respond at risk of simply dumbing down KF’s responses.

I don’t see how this get’s us to objectivity. If two mathematicians disagree about including the axiom of choice in the set theory, what objective standard should they refer to?

They can include or not include axioms as they wish. As long as they make clear which set of axioms their statements are respecting, and their statements actually respect those axioms, they’re both correct. Why fight?

As to whether or not to choose a particular set of axioms for some job…well, it depends on what you’re doing. If you’re modeling phenomena, you want a set of sets of axioms whose emergent theorems and structures best fit the empirical data and are most compatible with the best fitting sets of axioms for adjacent phenomena.

And if that isn’t objective…well, wouldn’t physics then not be objective?

21. 21
Bob O'H says:

kf @ 9 –

And so while there is room for creativity and surprise etc, we are not in the sort of arbitrary, whims and fancies situation that marks the subjective. Objective criteria obtain.

I think you’re mis-characterising the subjective. There are objective criteria in dance (dancers have to obey physics, for a start), but that doesn’t mean it’s not subjective. TBH, knowing mathematicians, I expect there are a lot of whims and fancies in mathematics.

22. 22
Bob O'H says:

LocalMinimum @ 20 – Amblyrhynchus was asking about mathematicians, so the real world isn’t involved. 🙂

More seriously, you’re right, that if they are just doing mathematics, then they can chose whatever axioms they want.

I suspect that even in physics there is a lot of subjectivity. There are often different ways of representing the same thing, and the choice of which to use can be subjective(*). Also, once one starts to model, there are usually a great many choices that can be made, without any objective criteria.

(*) there’s a really nice example in evolutionary biology. The theory behind the whole group selection debate was resolved after the selfish gene school and the group selection school had come up with their models, and then realised that they were actually different ways of writing the same model: they were just partitioning the equations in different ways. So the choice of which to use depends on which one you prefer, which is entirely subjective.

23. 23
Bob O'H says:

kf @ 11 – I haven’t read Wolfram’s book, so I wouldn’t want to comment on it. I’m not sure it’s really relevant, though. One way or another there still have to be axiomatic assumptions made.

24. 24
kairosfocus says:

BO’H:

the concept that one may choose axioms with utter personal freedom — just what whims and fancies is about — would indeed make for a discipline that would be utterly subjective. But that is exactly what is not true.

Axioms for a domain have to be coherent and sufficiently structured to be fruitful in laying out what I have been calling a logical model world — comparable to an architect’s framework for a building. They need to connect to the general body of mathematical knowledge, methods, principles and just plain facts; many of which antedate the spreading of axiomatisation in the grand wave across C19 into C20 that then imagined that sufficient frameworks of axioms would capture the full range of possibilities, only to be stopped cold by Godel c 1930.

At that point it was realised that complex bodies of mathematical thought would have in them facts that are accurate to reality but unreachable by sound, coherent — yes, there it is! — systems of axioms; where also no constructive process would deliver axiom sets guaranteed to be coherent. Thus the dream of absolute comprehensive truth in a grand scheme tied to in effect God’s DNA for reality collapsed.

But in its wake we see a clue of objective truth: facts that are so but are unreachable from a particular axiomatisation. The facts are therefore clearly not manufactured in toto by said axiomatisation.

Mathematics is irreducibly complex, in a second sense as just summed up: incompleteness of any coherent axiomatisation of systems comparably complex to arithmetic etc.

Mathematics, too will be objective, open ended and stands permanently humbled in its claims post 1930. (Not coincidentally, Physics — which separated from Math across C19 — was also humbled in a comparable period that ended c 1930.)

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 Ax1, Ax2 . . . Axn then perforce

Th1, Th2 . . . Thm.

Per Axi and Thj, Th k etc then also

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.

As to dancers dancing, they are constrained to be true to physical possibilities; much as are engineers. They may spin out dramatised stories and stunning displays, but such creativity is in the end not chaotic.

Engineers . . . a much more relevant discipline . . . have to conform to physical AND MATHEMATICAL/ LOGICAL/ QUANTITATIVE/ STRUCTURAL realities if their systems are to work, manifesting logical, dynamical, process and functional coherence. They cannot at whim set up fresh axioms willy nilly — and yes engineering is replete with mathematics, then use such in designs and hope they will work out just fine.

Similarly, computer programmers (and such programs are in effect model, logically driven worlds) must face GIGO.

Again, truth is accurate conformity to states of affairs of the coherent reality of physically actualised and abstract possible worlds. Objective truth is that which is accountable per sound principles and practices of warrant that credibly tie claims and inferences to things as they are per such states of affairs rather than whatever we may fancy.

I suspect, one problem is the definition of objective vs subjective, and have added a clip from the impressive Collins English Dictionary to the OP, they have a pattern of pithy, powerful definitions that I find very helpful. For ease of reference:

subjective
1. belonging to, proceeding from, or relating to the mind of the thinking subject and not the nature of the object being considered
2. of, relating to, or emanating from a person’s emotions, prejudices, etc: subjective views.

objective
1. (Philosophy) existing independently of perception or an individual’s conceptions: are there objective moral values?.
2. undistorted by emotion or personal bias

At deeper level, this hints of the Kantian ugly gulch problem.

Accordingly, I remind one and all that by the end of C19, the famed British Philosopher F H Bradley had already identified the critical incoherence in dichotomising an inner conscious phenomenological world of impressions and appearances from the external one of states of affairs and things in themselves:

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.]

I add, that there is no good reason to hold that external reality is and can only be physical; that is to import a materialistic, naturalistic perspective which may well be dominant but which (especially in evolutionary materialist form enabled by fellow travellers) is utterly, irretrievably self-referentially incoherent, reducing mindedness to the result of blindly mechanical and stochastic processes that are decisively undermining of responsible, rational freedom. And that is even without the challenge of accounting for the functionally specific complex organisation and information in the wetware computational substrates they would substitute for minded rational responsible contemplation.

As J B S Haldane put it:

“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.)]

In short, I suspect we are looking at controlling but self-referentially incoherent worldview notions that haunt our civilisation at this stage of decline.

KF

PS: That there are many examples in Math, Science, Engineering etc of diverse frameworks that converged on a common domain and often turned out to be substantially equivalent is not a demonstration of utter subjective freedom but instead of the silent but very real force of controlling reality. A good case in point is the original emergence of Calculus and the different schools of thought. Similarly, Quantum Mechanics is like that. In engineering it is common to have clusters of converging inventions emerge at about the same time too.

25. 25
Bob O'H says:

kf –

the concept that one may choose axioms with utter personal freedom — just what whims and fancies is about — would indeed make for a discipline that would be utterly subjective.

Indeed. And I’m not saying that. Mathematics has a subjective element, but at some point logic takes over. Engineering is, I’m sure, the same. Neither mathematics, engineering or dance is entirely objective or entirely subjective, but you will find evidence of both.

26. 26
kairosfocus says:

BO’H: I think fields are explored and bodies of knowledge are developed by subjects, often in the context of organisation or institutional power balances and wider cultural views and agendas. That said, we have constraints of warrant that lead to bodies of objective truth forming bodies of knowledge that are accountable before not just community power balances but also logic, facts and more broadly reality. Where that reality is not just physical but also abstract. This is where Mathematics (our current focus) draws its objective character from despite idiosyncracies, mistakes, power balances and more. Ironically, evolutionary materialist views actually undermine a requisite of being subjects, responsible, rational freedom capable of significant choice, including that to follow a chain of warrant and per insight, draw a conclusion. Reducing active rationally contemplative, morally governed [duties to truth, logic, prudence, fairness and more] mind to blindly mechanical and/or stochastic computational substrates is a big fail, even granting for argument that evolutionary narratives have probative force on getting to the required FSCO/I for such a substrate. KF

27. 27
Bob O'H says:

kf – I suspect we agree – maths (like almost everything else) is both objective and subjective.

Don’t worry, I won’t tell anyone.

28. 28
kairosfocus says:

BO’H: nice try. The issue is are knowledge claims, including axioms as integral parts of the framework, objective. to which the answer is clearly yes. That’s why Mathematics, in its settled part, is so routinely relied on and taken as a yardstick of solid objective truth. KF

29. 29
LocalMinimum says:

Bob O’H @ 22, 25:

I suspect that even in physics there is a lot of subjectivity. There are often different ways of representing the same thing, and the choice of which to use can be subjective(*). Also, once one starts to model, there are usually a great many choices that can be made, without any objective criteria.

The thought process of the individual is most certainly subjective; and so will be elements of their methodology. Agreed; though largely and specifically constrained if logic is obeyed. Also, we are left with choosing among models we know or build, and I’d be hard pressed to prove that society and culture hasn’t guided the order in which we discover/construct/favor models in some part at least.

As it is, I would think we both agree that the other party’s subjective sociopolitically emergent view of reality has affected perspectives on biology with some significant consequence.

However, if we consider all known, viable models of a subject as a part of a larger abstract structure, you’ll see that subjectivity is simply the order at which you pay attention to what parts of the greater whole; a choice that is necessitated by limits to computational resources. The models and math are persistent, outside of your attention or even knowledge of them; which makes them objective.

…and then realised that they were actually different ways of writing the same model: they were just partitioning the equations in different ways. So the choice of which to use depends on which one you prefer, which is entirely subjective.

And here it is. Choice among starting points and directions within the rules of the model is generally subjective; but the math of each model, being part of a larger objective abstract structure, led them to the connection. Even if the models ultimately prove incomplete or even wrong against the target phenomena with further data/analysis or better models, the discovered intersections are fact, and still yield precise, objective analogies with respect to frames of reference that don’t connect to (or are disconnected from) the contradictions.

30. 30
Bob O'H says:

Wow, kf. You won’t even acknowledge an agreement!

LocalMinimum – yes, in the case of group selection, both approaches turn out to be the same, but that isn’t always the case. And the data can’t always tell you which is the right model. Because as you point out, we can’t examine every model, subjectivity has to appear somewhere.

31. 31
StephenB says:

Attn: Kairosfocus

Bob, “subjective” refers to the subject who is investigating and “objective” refers to the object of the investigation. The principles of mathematics are the object of the investigation that is discovered by the subject, who is the investigator. They do not come from him but are discovered by him.

32. 32
EricMH says:

@ Bob then you do not believe mathematics is subjective.

33. 33
kairosfocus says:

BO’H: Kindly, scroll up to 18, where the focus is the core entities of Mathematics, numbers. You will find that there are several complementary approaches to the same core, and that these are facts to be accounted for by any general mathematical schema. A system of axioms that fails to address these adequately will fail, would not even make it to public notice. Going further, we have especially the irrationals and transcendental numbers, which were patently discovered rather than invented. In the case of irrationals, in classical times from the incommensurate relation of the sides and diagonals of a square; initially, recognised with horror. That reaction was subjective (and the pentagon was found to be much worse than the square) but it is a response to the objective facts and force of logic. This illustrates some of why I keep on stressing that axiomatisation is not a free willy nilly exercise, and why I keep on pointing out that new logic model worlds created through fresh axiomatisation will be connected back to the heart of mathematics and to facts in multiple ways that force an objective character. Yes, those who did not expect other geometrical domains were mistaken, buut that open-endedness is part of objectivity and — precisely because of the force of the warrant — the new developments were accepted. Likewise, set theory had to be reworked and there is no current simple, single formulation; that implies divergent branches on the various options. Also, when Godel showed that there are truths unreachable from axiomatisations, he underscored that there are realities independent of any particular axioms, i/e. the facts are not simply created by the axioms, though a good formulation can and does lead to fruitful discoveries. So, no, I cannot but see that the body of mathematical knowledge (including the axiomatisation approach) is objective. Being able to look at and approach a mountain from different angles which show different aspects, does not mean the mountain is a subjective jumble rather than a coherent whole. The story of the blind men and the elephant should give us pause. KF

34. 34
Bob O'H says:

EricMH – I believe that mathematics, in different respects, is both subjective and objective.

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.

35. 35
kairosfocus says:

BO’H:

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 that 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. In 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.

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 development of algebraic number theory in the 19th century and the proof of the modularity theorem in the 20th century. 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]

Notice 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.

KF