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\begin{document}
\title{On the logical consistency of special relativity theory and
non-Euclidean geometries: Platonism versus formalism}
\author{Radhakrishnan Srinivasan\thanks{R \& D Group,
IBM India Software Labs, 5th floor, Golden Enclave,
Airport Road, Bangalore~560017, India.
E-Mail:~\texttt{sradhakr@in.ibm.com}}}
\date{}
\maketitle

\begin{abstract}
The Lorentz transformations in the theory of special relativity~(SR) lead
to a little-investigated phenomenon called relativistic determinism.
When two relatively moving inertial observers A and B coincide in
space at a given instant, it is possible that a
particular distant event is in the future of one of the
observers~(B), but is in the present or even in the past
of the other~(A); this is a well-known consequence of the relativity
of simultaneity. Hence B's future at the instant of coincidence with A
is determined by the fact that A had already seen it at that instant.
In this paper, it is argued that Platonism
is inherent in relativistic determinism and
from the point of view of formalism, a logical inconsistency can
be deduced in SR, as formalized in classical first-order predicate
logic~(FOPL). Similarly, it is argued that Platonism is inherent
in non-Euclidean geometries~(NEG) and that formalism demands that Euclid's
fifth postulate~(EP) be provable in plane neutral geometry~(NG)
consisting of Tarski's axioms (as formalized in FOPL).
The essential argument here is that models
of NEG can only be constructed by assuming that the postulates
of Euclidean geometry~(EG) are metamathematically or Platonically `true'.
Formalism demands however that such Platonic truths do not exist and
so one concludes that formally, the provability of EP
follows from its truth in every model of NG.
The classical argument for `interpreting' NEG
within EG must be formally rejected as amounting
to assuming the Platonic/metamathematical truth of the Euclidean
postulates. So from the point of view of formalism, this
argument does not really prove the relative consistency of
NEG with respect to EG. An argument for provability of EP
in NG is presented in the non-Aristotelian finitary
logic~(NAFL) proposed by the author.
\end{abstract}

\section{Relativistic determinism -- the clash with logic}
Consider the theory of special relativity~(SR) as formalized in
classical first-order predicate logic~(FOPL). For details of such
a formalization, see the work of Andr\'eka \emph{et al.}~\cite{amn};
on page~1245, Part~VI of \cite{amn} an explanation is given
for why first-order logic is to be preferred to higher-order logics.
A particularly simple set of postulates for SR is given in~\cite{ah}
and we will adopt these for the elementary treatment in this paper.
The little-studied phenomenon of relativistic determinism, which is
a consequence of the Lorentz transformations and the relativity
of simultaneity, is clearly explained in~\cite{fk}. An example
of relativistic determinism which we wish to consider in this paper
may be formulated as follows.

Let A and B be relatively moving inertial observers who happen to coincide
in space at a given instant defined by $t=0$ in A's frame and
$t^{\prime} = 0$ in B's frame. Let $C$ be an instantaneous
event that is localized in space and distant to both A and B.
Let $U(IBC)$ define a non-trivial universe of
material objects with certain well-posed
initial-boundary conditions~$IBC$. Define the proposition $P$
as ``From A's point of view, $C$ occurs in
$U(IBC)$ when A's local clock reads $t=0$'' and the
proposition $Q$ as ``From B's point of view,
$C$ occurs in $U(IBC)$ when B's local clock reads
$t^{\prime}=T$''. Here $T>0$ is a constant obtained from the
Lorentz transformations as applied to the event $C$ in A's and B's
inertial frames. Relativistic determinism asserts that if $P$ is
true then $Q$ must be true (or $P \Rightarrow Q$); in other words,
B's future at time $t^{\prime}=0$ is determined by the fact
that A has observed $C$ at precisely that instant (when A and B
coincided) and so B must necessarily observe $C$ at $t^{\prime}=T$.

In order to obtain a logical contradiction from the above scenario,
let us further stipulate that the proposition
``Event $C$ occurs in $U(IBC)$'' is undecidable in SR, i.e.,
in particular, neither A nor B can either prove or refute this proposition.
Such undecidability could occur in many ways, for example, as a result of
G\"odel's incompleteness theorems; alternatively, $C$ could be a
probabilistic event, such as, the outcome of a coin toss experiment
or some quantum phenomenon; or else, $C$ could be completely
unpredictable as a result of being decided by the instantaneous free
will of a human being. It immediately follows that $P$ and $Q$ are
undecidable in SR; see the ensuing paragraph for the definition of
such undecidability.  Note however, that SR requires
$P \Leftrightarrow Q$ to be a theorem despite the undecidability of $P$
and $Q$; this fact immediately makes SR inconsistent in
the non-Aristotelian finitary logic~(NAFL) proposed by the author
in \cite{nafl} and \cite{nafl2} (in particular, see Remark~5 of
\cite{nafl} and Section~2.2 of \cite{nafl2}). This argument for
inconsistency of SR in NAFL is simpler than the one given using inertial
frames in \cite{ifr}. It follows that the philosophy of formalism
as embodied by NAFL~\cite{nafl2} immediately rejects relativistic
determinism. The goal of this paper is to show
that an inconsistency can be deduced in SR even
within FOPL, if one insists on formalism.

Henceforth, whenever we refer to A~(B), it is to be
understood that our argument may apply equally well to any observer in
A's~(B's) set of inertial frames. Note that we require the following
restrictions regarding propositions involving $P$ and $Q$. The
truth of $P$~($Q$) can be \emph{asserted} (via an
observation, for example) or \emph{deduced} in SR \emph{only} by A~(B).
However, B~(A) can consider and either accept or refute in SR any
assertion/deduction of the truth of $P$~($Q$) made by A~(B);
but B~(A) cannot \emph{assert} or \emph{deduce} the truth of
$P$~($Q$). The undecidability of $P$~($Q$) in SR means that A~(B)
can neither prove nor refute $P$~($Q$) in SR.  $P \Rightarrow Q$
is a theorem in B's (and not A's) frame; in other words, only B
has the right to deduce $Q$ in SR from an assertion of $P$ made
by A (if B happens to agree with A's assertion). Similarly,
$Q \Rightarrow P$ is a theorem in A's (and not B's) frame.
In fact $P \Rightarrow Q$ and $Q \Rightarrow P$ are illegitimate
propositions in A's and B's frames respectively. The idea behind
these restrictions is to allow A~(B) to consider the truth of
$Q$~($P$) without undermining the Lorentz transformations.

In particular, $Q$ is undecidable in SR, which
means, as noted above, that B can neither prove nor refute $Q$ in SR.
The question we wish to consider is as follows. Given that A has
asserted the truth of $P$, and given that $P \Rightarrow Q$ is a
theorem of SR in B's frame, can B accept A's assertion and conclude
$Q$? In the metatheorem that follows, we argue that B in fact has a
formal refutation of A's assertion; i.e., B has a
proof of $\neg P$ in SR and hence B has no way to
conclude $Q$ despite A's assertion of $P$. However,
B does not have the right to use  $Q \Rightarrow P$ along with
the said proof of $\neg P$ to deduce $\neg Q$, because, as noted
above, $Q \Rightarrow P$ is a theorem of A's (and not B's) frame.
Hence $Q$ continues to remain undecided in SR (in B's frame)
despite A's assertion of $P$. See Remark~\ref{rm6} below for
further clarifications.

Before proceeding to the main result in the metatheorem below,
we observe that an additional restriction is necessary, as follows.
A and B accept each other's observations/theorems as true/valid
if and only if there is no disagreement with (or a refutation of) the
observations or any step used in the proof of the said theorems,
including the theorems themselves. As an example, suppose A
asserts $\neg P$ and concludes $\neg Q$ from
the theorem $Q \Rightarrow P$ of A's frame. Then B accepts
A's assertion $\neg P$ as true and A's inference
$\neg P \Rightarrow \neg Q$ as valid despite that fact that
such an inference is illegal in B's frame. Thus B accepts A's
conclusion $\neg Q$ as true; i.e., B does not insist that
because of the illegality of the inference
$\neg P \Rightarrow \neg Q$ in B's frame, there must exist a
model for SR in which A asserts $\neg P$ and B asserts $Q$.

\begin{theorem}
Suppose \emph{A} claims the truth of $P$.
\emph{B} has a proof of $\neg P$ in \emph{SR}.
Formally, \emph{B} must accept this proof
rather than \emph{A}'s claim. Hence the theoremhood of
$P \Rightarrow Q$ does not decide $Q$ in \emph{SR}
from \emph{B}'s point of view. From the completeness
theorem of \emph{FOPL}, it follows that from \emph{B}'s
point of view, there must exist a model for \emph{SR} in which
$Q$ is false despite \emph{A}'s claim of the truth of $P$.
The existence of such a model would make \emph{SR} inconsistent
from \emph{A}'s point of view, because the Lorentz transformations would
be violated, and indeed, there could even be a disagreement between
\emph{A} and \emph{B} over whether $C$ occurred at all. If such a model
does not exist, then \emph{SR} is inconsistent from \emph{B}'s
point of view.
\end{theorem}
\begin{proof}
Define the proposition $R$ as
``From B's point of view, $C$ occurs in $U(IBC)$ when B's
local clock reads $t^{\prime}=0$''. Clearly, B has a proof of
$P \Rightarrow Q$ and hence a proof of
$P \Rightarrow \neg R$, from the Lorentz transformations;
B can only expect to observe $C$ at $t^{\prime}=T$ if $P$
is indeed true as claimed by A. It follows that if B
accepts A's claim of the truth of $P$, then B does indeed
have a proof of $\neg R$. Now $Q \& \neg R$ expresses the fact
that $C$ is in B's future when B's local clock reads $t^{\prime}=0$,
which means that from B's point of view, $C$ has not yet occurred
when A and B coincide with their local clocks reading $t=0$ and
$t^{\prime}=0$ respectively. To say that
$C$ has not yet occurred at this well-defined instant
(according to B's definition of simultaneity)
is also the same as making the global assertion that no one,
including any observer in A's frame, has observed $C$ at that instant
from B's point of view. It follows that B can conclude $\neg P$.
In other words, B is entitled to draw the inference
$Q \& \neg R \Rightarrow \neg P$. See Remark~\ref{rm3} below for
further justification of this inference. Since B has concluded
$\neg P$ starting from the assumption
$P$, it follows that B has a proof by contradiction of $\neg P$
in SR. Consequently, B concludes that $Q$ remains undecided in SR
despite A's claim of the truth of $P$ and the metatheorem follows.
\end{proof}
\begin{remark}\label{rm1}
At the well-defined instant when A and B coincided, defined by
$(t=0, \; t^{\prime}=0)$, the truth value of $P$ in A's frame is
classically determined; in other words, in A's frame, $P$ is either true
or false in FOPL at $t=0$ \emph{irrespective} of whether A
happens to \emph{know} the truth value at that instant.
If eventually A determines that $P$ is
true, it was \emph{already true} at the instant
$t=0$. So from B's point of view, A's claim of
$P$ already `exists' at the instant $t^{\prime}=0$ and by the
metatheorem there must exist a model for SR
in which subsequent events in B's frame (including actions of B
or anyone else in B's frame) falsify A's claim.
\end{remark}
\begin{remark}
Note that one could deny that $P$ is a
legitimate proposition in B's frame of reference. But then
B still has the problem of proving $Q$, given A's claim of $P$.
If B is not allowed to consider A's claim at all, then it follows
that B still has no proof of $Q$. In particular, for times
$t^{\prime}$ satisfying $0 \le t^{\prime} < T$, B still has no
proof of $Q$ and concludes that there must exist a model for SR in
which $Q$ is false, \emph{given} the data up to (and including) the
time $t^{\prime}$; B cannot consider that such data may include A's
claim of $P$, by assumption.  So formally, the conclusions
of the metatheorem would still follow. But if B were to abandon
formalism and accept the Platonic truth of $P$ (without formally
admitting $P$ as a proposition), then the Platonic truth of $Q$ would
also follow. In other words, one could argue that SR is about the
real world and $P$ and $Q$ are real-world truths. But even here
there is a problem -- if B were to accept the Platonic
(real-world) truth of $P$, B would have no option but to conclude
that the event $C$ ``really'' occurred when A and B coincided, and that
the truth of $Q$ is nothing but an illusion. One must remember that
it is only through formalism that such a conclusion was avoided in the
first place.
\end{remark}
\begin{remark}\label{rm3}
A second option might be to deny B's inference
$Q \& \neg R \Rightarrow \neg P$ made in the above proof.
But such a denial is not tenable, as explained below.
Note that there is a clear definition of global simultaneity in SR
(unlike general relativity) within an inertial frame of reference,
such as, that of B. So there is a clear past, present and future for
B and to say that an event $C$ can only occur in the future of B
at a given instant defined by $t^{\prime}=0$ is the same as saying
(from B's point of view) that $C$ can also only occur in
the future of A at this instant, when A and B coincided. This
immediately implies an assertion of $\neg P$ by B, for at the
instant of coincidence, B concludes that A still does not have
any proof on whether $C$ will occur at all in the future; this
is especially clear if $C$ is a probabilistic event, such as, the
outcome `heads' in a coin toss experiment, as considered in the following
remark. It is very important to note that the converse does not apply from
A's point of view. That is, given that B has observed $Q$ to be true,
A concludes $P$ from the Lorentz transformations. But A cannot infer
$\neg Q$ from $P$ (in a manner similar to B deducing $\neg P$ from $Q$),
despite the temptation to do so. A can only conclude that $P$ is true
and that the local observer in  B's frame has observed that event $C$
has occurred at the instant $t=0$ in A's frame, but B has the
\emph{illusion} of observing $C$ at a later time; hence, from A's
point of view, B accepts the truth of $Q$ due to a wrong definition
of simultaneity. But there is no scope for B to argue in this manner
in order to deny the inference $Q \& \neg R \Rightarrow \neg P$; there
is no way for A to get the illusion of an event $C$ that has not yet
happened, and indeed, \emph{need not happen at all}, from B's point of view.
This asymmetry clearly highlights the problem with relativistic
determinism.
\end{remark}
\begin{remark}
Let $C$ be the probabilistic outcome `heads' in a coin toss
experiment. Suppose A claims $P$. From B's point of
view, the event $C$ had not yet occurred at $t^{\prime}=0$
on B's local clock and so B concludes that A's frame still
did not have any evidence of the outcome `heads' at the instant
when they coincided. Hence from B's point of view, at the instant of
coincidence with A, A's frame has no information available on whether
$C$ will occur at all. B deduces that it certainly cannot be A's point
of view that $C$ occurred at $t=0$ on A's local clock; it follows
that B has a refutation of A's claim of $P$ and concludes
that either outcome (`heads' or `tails') is still possible
for all times in the interval $0 \le t^{\prime} < T$.
In particular, since B allows for the outcome `tails', there
is no way (from B's point of view) that A got the illusion of
`heads' via a wrong definition of simultaneity. Conversely, from B's
claim of $Q$, A concludes $P$ and that the local observer in B's frame
had already seen the outcome `heads' at $t=0$, when A and B coincided.
So from A's point of view, B's frame already has evidence of `heads'
at this instant, although B may not yet be aware of it.
Hence A cannot refute B's claim of $Q$, as noted earlier.
\end{remark}
\begin{remark}
Consider the following example. Let A be at the front end of a
long platform at rest and let B be at the front
end of a long train.
The train is adjacent to the platform and is travelling at a
very high, constant velocity (close to the speed of light) relative
to the platform. Define the event
$C$ as the (instantaneous, localized) coincidence
of the rear end of the train with the rear end
of the platform. Let A and B coincide in space at an instant defined
by  $t=0$ and $t^{\prime}=0$ respectively on their local clocks. Define
$P$ and $Q$ as before. Suppose that
there is a localized fault in the track
just prior to the rear end of the platform, so that it is completely
unpredictable as to whether the wheels of the
train will instantaneously derail or pass smoothly over this fault.
It follows that $P$ and $Q$ are undecidable
in SR. Suppose A claims that $P$ is true;
i.e., A determines that no derailment has occurred.
From the metatheorem, B has no proof of $Q$ in SR and concludes that
there must exist a model for SR in which $Q$ is false
despite A's claim. In particular, B
allows for the possibility of derailment
of the train at the fault at any time $0 \le t^{\prime} < T$ despite
A's claim to the contrary and this would cause $Q$ to be false.
As noted in Remark~\ref{rm1}, at ($t=0, \; t^{\prime}=0)$, neither
B nor A need actually be aware of A's (eventual) claim
in order for this argument to apply.
Suppose one insists that SR works in
the real world and that B would always find $Q$ to be true if A claims
the truth of $P$. Let there be a non-zero probability $p < 1$ such
that derailment occurs in the time interval
$0 \le t^{\prime} < T$, from B's point of view. In a series of
experiments in which A claims $P$, B would, by hypothesis,
find $Q$ to be true. Since B has a formal refutation of $P$ in SR
(by the metatheorem), it follows that B will conclude
that the laws of probability have been violated in the real world.
Perhaps this argument would throw some insight into why SR may
be incompatible with quantum mechanics. 
\end{remark}
\begin{remark}\label{rm6}
The restriction that $P \Rightarrow Q$~($Q \Rightarrow P$) can
only be a theorem in B's~(A's) frame may seem somewhat artificial
to the reader. Observe that if $Q \Rightarrow P$ were to be a theorem
in B's frame, then B's proof of $\neg P$ would immediately
yield a proof of $\neg Q$, which implies a complete rejection
of the Lorentz transformations and SR. Indeed, B's proof of $\neg P$
simply expresses that B disagrees with A's claim of $P$, which is within
the spirit of SR; this proof should not be allowed to enable B
to conclude $\neg Q$, thus contradicting SR. On the other hand,
since SR requires B to refute A's claim of $P$, such a claim
cannot be used by B to prove $Q$ either. This is the main point of
this section, namely, that $Q$ continues to remain undecided in SR
from B's point of view despite A's claim of $P$.
Note that since $Q \Rightarrow P$ is
a theorem in A's frame, A is free to conclude $\neg Q$ from
an assertion of $\neg P$ in A's frame. Here B would have no
problem in accepting A's conclusion of $\neg Q$ as true
because B does agree with any claim of $\neg P$ by A,
and B accepts the validity of A's inference
$\neg P \Rightarrow \neg Q$ despite its illegality in B's frame.
\end{remark}
\begin{remark}
Note that the existence~(non-existence) of the model stated in the
metatheorem makes SR inconsistent from A's~(B's) point of view.
Similarly, B's proof of $\neg P$ will not be accepted by A as
correct. This raises serious philosophical issues
of who decides the consistency of a theory and
the validity of a proof within that theory. The author's
opinion is that these issues can be settled
only by agreement amongst the entire human race, irrespective of
the frame of reference any particular individual happens to be in.
In this respect, SR seems to be an illegitimately formulated
theory. To assert that there is a reality for the consistency
of SR independent of (and possibly contrary to) the deductions of
human beings made from the inertial reference frames they happen to
be in, seems to be highly questionable. It is clear that SR does
not tolerate undecidable propositions of the type required in
this paper. This seems to clash with G\"odel's incompleteness
theorems and rules out consideration of any probabilistic,
spatially localized events in SR.
\end{remark}

\section{Non-Euclidean geometries}
It was noted in \cite{nafl2} that Platonism is inherent in
classical logic and that the author's proposed non-Aristotelian
finitary logic~(NAFL) is the only logic that correctly embodies
formalism. Here we will first argue the inconsistency of
non-Euclidean geometries from the point of view of NAFL. We
will then critically examine non-Euclidean geometries in classical
first-order predicate logic~(FOPL) and explain precisely why
Platonism is inherent in these geometries, which must therefore
be rejected from the point of view of formalism.

In this paper we confine ourselves (unless otherwise
indicated) to plane geometry; the extension to the
three-dimensional case is straightforward. We assume that the
reader is familiar with the axiom scheme for plane
Euclidean geometry given in FOPL by Tarski~\cite{at}.
We use EG, HG and NG to denote Euclidean, hyperbolic and neutral
geometries respectively. Let $\psi$ be Euclid's fifth postulate;
then $\mbox{EG} = \mbox{NG} + \psi$, and
$\mbox{HG} = \mbox{NG} + \neg \psi$. We also assume that
the reader is familiar with Euclid's original formulation
in terms of his five postulates -- an excellent elementary
account is given by Greenberg~\cite{mjg} who also presents
Hilbert's (second-order) axiomatization of Euclidean geometry.
A good web reference is due to Royster~\cite{dcr}.
Our analysis applies equally well to Hilbert's formulation as
well, except that in second-order logic there is no completeness
theorem (which we will later require to argue that $\psi$
must be provable in NG).

\subsection{Inconsistency of non-Euclidean geometries in NAFL}
In the logic NAFL proposed by the author~\cite{nafl, nafl2}
the Main Postulate asserts that an undecidable proposition
$\phi$ in a consistent NAFL theory T (which has the same
rules of inference as in FOPL) can be true~(false) with respect
to T if and only if $\phi$ is provable~(refutable) in an
interpretation T* of T. Here T* is also an axiomatic NAFL
theory which, like T, resides in the human mind. Provability/refutability
of $\phi$ in T* is essentially equivalent in NAFL to an axiomatic
declaration of truth/falsity of $\phi$ with respect to T.
There is no Platonic world in which $\phi$ can be true or false
independent of axiomatic theories and independent
of an axiomatic declaration of such truth/falsity made
in the human mind via T*. Metatheorems~1~and~2 of \cite{nafl} explain
why the laws of the excluded middle and non-contradiction must
fail in the absence of such an axiomatic declaration
(i.e., when $\phi$ is undecidable in T*); in particular,
metatheorem~2 asserts that $\phi$ is neither true nor false
with respect to T in this case, which corresponds to a
non-classical model for T in which $\phi \& \neg \phi$ is
the case. Hence consistency of T demands the existence of
such a non-classical model. Here T* is the
`truth-maker' for a model of T, wherein only the theorems of
T* are assigned `true'; every other proposition
is in a superposed state of `neither true nor false'.
Note that `$\phi$' in this superposed state of $\phi \& \neg \phi$
in the non-classical model is to be interpreted as
`$\neg \phi$ is not provable in T*' and `$\neg \phi$' is
to be interpreted as `$\phi$ is not provable in T*'. This
interpretation is obviously true and so there is no contradiction
in the superposition required by NAFL. An important consequence of
the above truth definition is that the superposition
of any two models for an NAFL theory T must
also be a (possibly non-classical) model for T --
here the model is is to be understood as `non-classical' with
respect to propositions that are in a superposed state, and
`classical' with respect to other propositions.

A second important consequence of the NAFL truth definition
which we will require here is as follows. Suppose an NAFL theory
T requires a certain object (such as, `line' in Euclid's postulates)
to be uniquely defined in every model for T. Then consistency of
T demands that T must necessarily provide a unique \emph{construction}
(or \emph{definition}) for that object. In other words,
`non-constructive existence' of such uniquely defined objects is not
permitted; NAFL requires uniqueness to be enforced with respect to
the \emph{theory} T, and not just in \emph{models} for T.
For example, take T to be Euclid's first four postulates.
Classically, one can get a Euclidean model~E for T by interpreting
`line' to mean Euclidean straight line, and a hyperbolic model~H
for T by interpreting `line' in the hyperbolic sense. In
NAFL, the consistency of T demands that the superposition of
these two models H and E also be a non-classical model for T in
which $\psi$ is neither true nor false, as noted in the previous
paragraph. But such a superposed state of $\psi \& \neg \psi$
will violate the requirement of the first postulate of T (i.e., Euclid's
first postulate) that a `line' be uniquely defined by
any two of its distinct points; it follows that
the required non-classical model for T cannot exist.
Thus we conclude that the undecidability of $\psi$ in T makes T
inconsistent in NAFL.

To sum up, it is inconsistent in NAFL to assert that
T permits both Euclidean and hyperbolic definitions of a straight line,
given that T requires straight lines to be uniquely defined
by any two of its points; such an assertion clearly
implies a lack of uniqueness \emph{with respect to} T. It follows that
NAFL does not permit entities like `point', `line', `plane', etc.\ to
be left uninterpreted (or non-constructively defined) in T
because of the fact that these entities must have a unique
construction available in every model for T. For example, T requires
that any given object is either a point or not a point; in
set-theoretical terms, the class of all points (which is precisely
the `plane') unavoidably exists~\cite{nafl2} in the NAFL
version of T and the axiom of extensionality for classes
will require that a given object either belongs to or does not
belong to that class. Therefore consistency of T in NAFL demands that
two classical models for T, in which a given entity is a point in one
of the models and not a point in the other,
cannot both exist; for the superposition of such classical models
cannot be a non-classical model for T as required.
It follows that the NAFL version of T does not tolerate any ambiguity
in the meaning of `point'. Similarly, all classically `uninterpreted'
entities must necessarily be constructively defined in the NAFL
version of T (in fact, to have their Euclidean meanings, as we will
argue shortly).

Another simple argument for the decidability of $\psi$ in the NAFL
version of NG is as follows. Suppose, to get a contradiction, that
$\psi$ is undecidable in NG. The Saccheri-Legendre theorem of NG
(see Chapter~4, pg.~101 of \cite{mjg}) asserts
that the sum~$S$ of the degree
measures of the three angles in any triangle is less than
or equal to $180^{\circ}$. Consider the proposition $\Psi$ defined
by `$S=180^{\circ}$', with the negation $\neg \Psi$ taken as
`$S<180^{\circ}$'. It is easy to show that
$\Psi \Leftrightarrow \psi$ and hence by hypothesis, $\Psi$ is
undecidable in NG; $\Psi$ corresponds to EG and $\neg \Psi$, to HG.
In NAFL, the consistency of NG and the assumed undecidability
of $\Psi$ demands that there exist a non-classical model for NG in which
$\Psi \& \neg \Psi$ is the case. But it is also a theorem of
NG that $S$ be uniquely defined, and so formally, the superposed
state of $\Psi \& \neg \Psi$ violates this uniqueness requirement.
It follows that the required non-classical model for NG cannot exist
and so the NAFL version of NG would be inconsistent if $\psi$ were
to be undecidable as assumed. The conclusion is that in NAFL, the
rules of inference of NG must necessarily be such that $\psi$ be
either provable or refutable; under the ensuing heading, we will
argue the case for provability.

\subsubsection{Proof of $\psi$ in the NAFL version of NG}
Classically, consistency of NG demands undecidability of
$\psi$ in NG; this is diametrically opposite to consistency in NAFL, which
demands \emph{decidability} of $\psi$ in NG, as noted above.
In the ensuing subsection we will demonstrate
that the classical argument is valid if and only if one accepts
Platonism (which implies a rejection of NAFL).

In order to argue for the provability of $\psi$ in NG,
we first note that `point', `line', `plane' and other
classically uninterpreted entities of NG must necessarily have
unique, constructive definitions as demanded by NAFL.
Our contention is that it is precisely the
addition of $\psi$ to NG which provides
such unique, constructive definitions; $\psi$,
together with the axioms of NG, are \emph{essential}
in order to have a meaningful NAFL theory.
Hence the axioms of EG cannot be denied in NAFL and must be
declared as tautologously true. At this stage the reader might
wonder why $\psi$ is essential; why not $\neg \psi$?
Note that $\neg \psi$ still does not
provide $\emph{unique}$ meanings to the
uninterpreted terms; there are many possible classical
interpretations of HG, such as, the Beltrami-Klein model,
Poincar\'e's models, etc.; see Chapter~7 of \cite{mjg}.
This ambiguity is not acceptable in NAFL because the superposition
of these classical models of HG cannot be a (non-classical) model
for HG, as demanded by NAFL; it is only $\psi$ that removes all
ambiguities. In the ensuing subsection, we argue that Platonism
is inherent in $\neg \psi$, which must be rejected by NAFL.
The problem we are faced with is how the definitions
and rules of inference of classical NG must be modified so that
a proof of $\psi$ results in the NAFL version (which, as noted
earlier, uses the classical rules of inference). If such a
modification is deemed impossible, then NG is inconsistent in NAFL.
An attempt at a solution follows.

Playfair's postulate, which is equivalent to Euclid's
fifth postulate $\psi$, asserts~(\cite{mjg}, Chapter~1, pg.~17)
that for every line $l$ and every point P that does
not lie on $l$, there exists a unique line $m$ through P that
is parallel to $l$. Henceforth, we will refer to Playfair's postulate
as $\psi$. Classically, two lines are parallel if and only if
they do not intersect. But this definition is not satisfactory
in NAFL as it leads to undecidability of $\psi$ in NG and the
ambiguities in the uninterpreted terms noted above. The
NAFL definition of `parallel' is stated as follows.
\begin{definition}\label{df1}
Two distinct coplanar lines are parallel if and only if they are
equidistant at all points, where distance between the lines at a
point (on either line) is defined as the length of the
perpendicular to the other line dropped from that point.
Similarly, a line segment AB that does not lie on a line $l$ is
parallel to $l$ if and only if AB is equidistant from $l$ at
every point of AB.
\end{definition}
Definition~\ref{df1} is in fact first due to Posidonius as the
following quote from (\cite{dcr}, ``The Origins of Geometry'') shows:
\begin{quote}
``Many people have tried to prove the Fifth Postulate. The first
known attempt to prove Euclid V, as it became known,
was by Posidonius (1st century B.C.). He proposed to
replace the definition of parallel lines (those that do not
intersect) by defining them as coplanar lines that are
everywhere equidistant from one another. It turns out that
without Euclid V you cannot prove that such lines exist.''
\end{quote}
A similar definition was also used later by
Geminus~(10~B.C.\,--\,$\sim$\,60~A.D.) in a failed attempt to prove
Euclid's fifth postulate from the first four; see the quote below
from \cite{ocr}:
\begin{quote}
``Geminus tried the following approach giving a definition of
parallel lines:-\\ \\
\emph{Parallel straight lines are straight lines situated in
the same plane and such that the distance between them, if they
are produced without limit in both directions at the same time
is everywhere the same}.\\ \\
The `proof' which Geminus then gave of the parallel postulate
is ingenious but it is false. He made an error right at the start
of his argument for he assumed that the locus of points at a fixed
distance from a straight line is itself a
straight line and this cannot be proved without a further postulate.
It is interesting, however, that Geminus attempts to prove the parallel
postulate and, although it is unlikely to be the first such attempt,
at least it is the earliest one for which details have survived.''
\end{quote}
Of course, we do not wish to repeat the mistakes of these attempts.
\begin{proposition}\label{pr1}
Given a line $l$ and a point \emph{P} at an arbitrary non-zero distance
$D$ from $l$, there exists a unique line segment \emph{M}
through \emph{P} parallel to $l$, such that \emph{P} is
at the midpoint of \emph{M} and \emph{M} is of a given arbitrary non-zero
length $L$. Here $D$ and $L$ are (standard) finite lengths.
The line segment \emph{M} will remain parallel to
$l$ when extended by an arbitrary (standard) finite length
such that \emph{P} continues to remain at the midpoint of M.
Here `parallel' is defined in Definition~\ref{df1}.
\end{proposition}
Note that Euclid's second postulate, which is provable in NG,
permits the extensions noted in Proposition~\ref{pr1}. A proof
of Proposition~\ref{pr1} is, of course, impossible in the
classical version of NG. In the spirit of Euclid,
we will permit `reasoning from diagrams' as a rule of inference
added to those of classical NG, in order to overcome the above
difficulty; call the resulting theory NG(NAFL), in which the
uninterpreted entities, such as, `point', `line' and `plane',
are restricted to necessarily have their Euclidean meanings
in any diagrammatic proof. In this paper, we are only concerned
with how NG(NAFL) handles $\psi$; the question of how the axioms
and rules of inference of NG must be modified in NAFL to handle a
continuum of real numbers (if at all it is possible) is reserved
for future work.
\begin{proof}[Proof of Proposition~\ref{pr1} in \emph{NG(NAFL)}]
We depict the line $l$ on the diagram as a (sufficiently long,
Euclidean) line segment with arrowheads at the
end-points pointing outwards (i.e., away from
the center of the line segment). Given the point P, the unique (Euclidean)
line segment M is then constructed as in Proposition~\ref{pr1}, using
a protractor and ruler, after appropriately scaling down (or scaling
up) the length~$L$ of M and the distance~$D$ of P from $l$;
the scale factors for these two scalings in mutually
perpendicular directions need not be the same. Note that
the line segments of $l$ are also scaled by an identical factor to
that of M. We claim that this would be a diagrammatic
`proof' of Proposition~\ref{pr1} in NG(NAFL). Two diagrams suffice
for this `proof', with P on either side of $l$; any change
in $L$ or $D$ would merely imply a change in the scale factors
of the diagrams.
\end{proof}
\begin{remark}
Scaling down (scaling up) the length of a line segment
amounts to translating a long~(short) line segment into a
shorter~(longer) one; since there is a one-to-one
correspondence between the points of any two
such line segments, the said translation is a legitimate proof
technique that must be `wired' into the rules of inference.
Does such a `proof' presume $\psi$ and Euclidean concepts?
Probably, but note that real-life diagrams cannot in any sense be equated
with the ideal continuum concepts embodied in the axioms of NG.
One should simply view this real-life construction as a mechanical
procedure that establishes the desired result.
Secondly, we will only need a finite construction of the line segment~M
on these diagrams which can be carried out with a ruler and a protractor;
but $\psi$ requires an infinite construction of the line $m$.
The fact is that the truth of Proposition~\ref{pr1} can be
indisputably depicted in these diagrams and the only way to `prove'
it without explicitly invoking $\psi$ is to incorporate this
fact into the classical rules of inference. This is perhaps
not a very desirable state of affairs, but in the absence of
alternatives, we will have to accept it. We might rationalize
that this diagrammatic `proof' is simply another way of asserting
Proposition~\ref{pr1} as a tautology in NAFL, i.e., it cannot
be denied. The diagrams express the Euclidean construction we must
unavoidably have in mind, but cannot express in the language of
NG without $\psi$ (or its equivalents), when we think of `line' or
`line segment'. See the ensuing subsection for why such a
Euclidean construction is unavoidable.
\end{remark}
\begin{proof}[Proof of $\psi$ in \emph{NG(NAFL)}]
Since $L$ and $D$ are \emph{arbitrary} constants, we claim that
Proposition~\ref{pr1} provides a \emph{direct, constructive} proof
of $\psi$ in NG(NAFL). The reader may balk at this assertion;
after all, is not M a \emph{line segment} of \emph{finite} length~$L$,
rather than the infinite line $m$ demanded by $\psi$?
The answer is surprisingly simple. See Sec.~2.2, pg.~14 of \cite{nafl2},
under the heading ``Open formulas and the meaning of `existence' in
NAFL'', where it is explained that in NAFL, open formulas (with a free
variable) or formulas with an `arbitrary' constant (such as,
$L$ and $D$ above) are in fact universally quantified
formulas with respect to the said variable
or constant. This is so because the values of
$L$ and $D$, being undecidable and unspecified in NG(NAFL), must
be in a superposed state of assuming all possible values. In particular,
Proposition~\ref{pr1} is automatically quantified over all possible
\emph{standard} values of $L$ and $D$. Since there are no nonstandard
models of arithmetic (and hence, of NG) in NAFL~\cite{nafl2},
it follows that such quantification is universal and immediately
implies a proof of $\psi$ in NG(NAFL) as explained below.

The uniqueness of the line $m$ as required by $\psi$ follows
from the fact that in NAFL, a `line' must be considered as a
`potential' rather than an `actual' infinity. The superposed
state of M with all possible (standard)~values of the length
$L$ \emph{is} the line $m$ in NAFL. Note that this interpretation
requires the universal quantification to be of the form
\[
\forall D \: \forall L \: \mbox{Proposition}\ref{pr1},
\]
with the quantifier for $D$ being outermost. Thus for each $D$, $L$
is in a superposed state of assuming all possible values.
The line $m$ is to be interpreted as an infinite class consisting
of the union of all possible line segments in the above formula,
with each line segment M in the union identified uniquely by its
given standard finite length $L$ (as defined in
Proposition~\ref{pr1}). Note that $m$ may be
represented by the union of any divergent, strictly increasing sequence
of standard finite lengths $\{L_1, L_2, L_3, \dots\}$ of
the segment M; every point of $m$ is a point of some segment of
length $L_j, \; j \ge 1$, in this sequence and the converse
also holds. The uniqueness of $m$ immediately follows;
if $m$ and $m_1$ are two lines that are obtained from this
construction, every point of $m$ is a point of $m_1$ and vice versa.
An infinite class is not a mathematical object
in NAFL~\cite{nafl, nafl2}. The axiom of extensionality
for classes states that a class is identified uniquely by its
elements; so the existence and uniqueness of each element M
(of given length $L$) of the infinite class
$m$ ensures that $m$ itself exists uniquely.
\end{proof}
\begin{remark}
Suppose one starts with $L=L_0$, i.e., a fixed segment M of
length~$L_0$ (a pure number) in the above proof. From Euclid's
Postulate~II~(\cite{mjg}, Chapter~1) one concludes that
this segment can be extended (such that P continues to remain
at the midpoint of M) to lengths $L=nL_0$, where
$n=2,3,4,\dots$, i.e., for all \emph{standard}
positive values of the integer $n$. In NAFL,
nonstandard models of arithmetic do not exist~\cite{nafl2},
and so the formula asserting the existence of the segment M of length
$L=nL_0$ is universally quantified over all positive (standard)
integers $n$. This amounts to a construction of the line $m$.
\end{remark}
\begin{remark}\label{rm10}
The classical objection to the above proof might be that the
diagrammatic construction of the line segment M
is only possible for \emph{standard} values of the lengths
$L$ and $D$. In FOPL, Proposition~\ref{pr1} is \emph{not}
to be treated as universally quantified; each different
(standard) value of $L$ and $D$ corresponds to a different
formula which requires a different proof. FOPL, unlike NAFL,
maintains a distinction between `arbitrary
but fixed' constants like $L$ and $D$, and a free variable.
There is no way to express `standard finite' in weak FOPL theories
in which Tarski's axioms for NG may be formalized; note that
Hilbert's axioms are not in FOPL, but in second-order logic
(or many-sorted logic) and do not admit
nonstandard models. So Proposition~\ref{pr1} is really
a proposition \emph{scheme} in FOPL, consisting of infinitely many
instances of the values of `fixed constants'
like $L$ and $D$, which are required to be standard
finite. The classical argument presumably is that
the diagrammatic construction of M does not take into account the
existence of nonstandard models for NG, in which $L$ and/or $D$ have
nonstandard values and in which $\psi$ is possibly false. In NAFL,
however, `standard' is a superfluous predicate~\cite{nafl2} and
Proposition~\ref{pr1} is indeed a legitimate proposition that
is universally quantified as noted in the above proof.
\end{remark}
\begin{remark}\label{rm11}
Consider a specific line~$l$ and a specific point~P at a
fixed distance $D$ from $l$. The \emph{direct, constructive}
proof of $\psi$ in NG(NAFL) given above fails in FOPL, because
such a `proof' would be infinitely long, as noted in
Remark~\ref{rm10}. Here we have kept P and $D$ fixed, but nevertheless
the `proof' would have to cover infinitely many instances
of the length $L$ in order to establish a construction for the
line $m$, and would therefore be no proof at all in FOPL.
This is the same as saying that $\psi$ cannot be established
in NG by directly extending a Euclidean line segment~M through
arbitrarily large standard finite values, because such a
construction does not prove the existence of $m$, which is
infinitely long. For a very simple analogy, note that it
would be wrong in FOPL to infer the existence
of an infinite set or class of natural numbers~(real numbers)
from the existence of infinitely many natural numbers~(real
numbers); a separate axiom would be needed to establish
the set or class in question. In NAFL, however, the existence
of infinitely many natural numbers immediately establishes the
existence of the infinite (proper) class $N$~\cite{nafl,nafl2};
a line is similarly  modeled as the union of a proper class
of infinitely many Euclidean line segments as noted earlier.
\end{remark}
\begin{remark}
By the completeness theorem of FOPL, one would expect that
there must exist a nonstandard model for NG in which
Proposition~\ref{pr1} (with Euclidean concepts) is true for the
specific line~$l$ and point~P of Remark~\ref{rm11}, but $\psi$ fails.
The failure of parallelism (as defined by Definition~\ref{df1})
in this model should only appear at nonstandardly long distances
from the point P, where Proposition~\ref{pr1} does not apply;
these distances are formally classified as `nonstandard
finite' but are `really' infinite. What is extremely
surprising (to the author at least) is that such a
nonstandard model cannot exist. This is so because
\emph{each instance} of Proposition~\ref{pr1} with
Euclidean concepts provides an \emph{indirect}
proof of $\psi$ in NG\,! To see this, take one particular
instance in which the (Euclidean) line segment~M parallel to $l$
has been constructed for specific, standard values of $L$ and $D$. Drop
perpendiculars from the end-points of $M$ to the line $l$ and consider
the rectangle bounded by $M$, $l$ and these perpendiculars. The very
existence of such a rectangle, whose angle sum is four right angles,
is equivalent to and proves $\psi$ in
NG~(\cite{dcr}, ``The Origins of Geometry''). It is very odd
indeed that on the one hand,  infinitely many instances of
the truth of Proposition~\ref{pr1} (with Euclidean concepts)
do not prove $\psi$ in NG; on the other hand, each such
instance of Proposition~\ref{pr1} does, after all,
prove $\psi$ in NG\,! One can only conclude that in FOPL, it would be
inconsistent to insist that only Euclidean meanings must be retained
for the `uninterpreted' terms of NG; consistency demands that
non-Euclidean meanings must necessarily be admitted, and so the indirect
diagrammatic proof of $\psi$ would be invalid in NG (as would the
diagrammatic proof of Proposition~\ref{pr1}). This is tantamount
to insisting that `line' must \emph{necessarily} have a
non-constructive existence in NG; it would be
impossible to prove in NG that parallel lines even exist, with
`parallel' defined as in Definition~\ref{df1}. This is diametrically
opposite to consistency in NAFL, which demands that \emph{only}
constructive Euclidean concepts must be admitted in NG(NAFL);
that `parallel' must necessarily be defined as in
Definition~\ref{df1}; and that $\psi$ must necessarily be
provable in NG(NAFL). The author believes
that the NAFL position is not only the logically
consistent one from the point of view of formalism,
but is also more natural; the diagrammatic
proof in NG(NAFL) directly confirms our intuition that
Proposition~\ref{pr1} is true in the real world.
\end{remark}

\subsection{Inherent Platonism in non-Euclidean models of NG}
The main thesis of this subsection is that non-Euclidean models
of NG can only be constructed by assuming the Platonic/metamathematical
truth of the postulates of Euclidean geometry~(EG).
This, of course, is objectionable from the point of view of
formalism even in FOPL. In NAFL, an outright contradiction can
be deduced as follows. Truth for the postulates of EG must necessarily
be \emph{axiomatic} in NAFL; there is no Platonic world in which
the Euclidean postulates are `really' true. So one has
\emph{axiomatically} declared EG to be true and then `re-interpreted'
terms like `point', `line', `plane', etc.\ into their non-Euclidean
meanings in order to generate the non-Euclidean model.
The axiomatic nature of NAFL truth clearly does not permit
such `re-interpretation' of Euclidean objects
into non-Euclidean ones, for once we have axiomatically declared
EG to be true, there is no scope for any change in the Euclidean
meanings of terms which are classically deemed `uninterpreted'.
Yet classically, it is precisely such an argument that is used
to prove the relative consistency of non-Euclidean geometries
with respect to EG; see Chapter~7 of \cite{mjg}.

In particular, let us consider the example of the Beltrami-Klein~(BK)
model of hyperbolic geometry~(HG) discussed in Chapter~7 of \cite{mjg}.
First EG is assumed to be Platonically `true' and a circle $\gamma$
of Euclidean radius $r$ is constructed in the Euclidean plane.
The hyperbolic plane is then defined as the interior of $\gamma$.
A chord of $\gamma$ is a segment AB joining two points A and
B on $\gamma$. The segment without its end-points A
and B is called an open chord. Hyperbolic lines are defined as
open chords of $\gamma$, with hyperbolic points retaining the same
meaning as Euclidean ones. After similar re-interpretation
of other `uninterpreted' terms of NG from their Euclidean meanings,
the axioms of HG thus obtained are `translated'
back into their Euclidean counterparts
and proved in EG in order to establish the relative consistency of
HG with respect to EG (and thereby the undecidability of $\psi$ in NG,
assuming EG to be consistent).

From the point of view of NG(NAFL), the construction of the
BK~model \emph{explicitly} and illegally assumes the Platonic truth
of $\psi$. To see this, note that the radius of $\gamma$ is of
\emph{arbitrary} (Euclidean) length. Consider two parallel chords of
equal length $l_*$ in $\gamma$, where `parallel' is in the sense of
Definition~\ref{df1}. Clearly, $l_*$ can be assigned an
arbitrary (but constant) value, given that the radius
of $\gamma$ is also arbitrary. In NG(NAFL) this amounts to
an explicit, constructive validation of $\psi$, as noted in
Proposition~\ref{pr1} and its proof. It is also clear that
the definitions of various terms (such as, `length') and proofs of
propositions of HG in the BK~interpretation use constructions of points
on $\gamma$ and even points and line segments outside of
$\gamma$~\cite{mjg}; these constructions have only Euclidean
meanings and so must be assumed to `really' exist. In other words,
$\gamma$ and its exterior in the Euclidean plane must necessarily exist
Platonically in order to define terms and execute proofs of HG in the
BK~interpretation. Such Platonic existence is illegal in NG(NAFL)
and hence the BK~model does not exist from the point of view
of NAFL. In fact NAFL predicts that the same situation holds of every
conceivable candidate for a model of HG because $\psi$ is provable
in NG(NAFL), as noted in the previous subsection. The axioms
of EG must necessarily be `really' true in every model of NG
even from the point of view of FOPL; the completeness theorem
demands that $\psi$ be provable in NG and this makes NG
inconsistent in FOPL, if one insists on formalism. However, it
has been demonstrated in \cite{nafl2} that formalism is not
a valid philosophy of FOPL.

Can the above situation be generalized to hold of \emph{any}
non-Euclidean geometry, whether two- or three-dimensional?
Again, NAFL predicts that this is the case, i.e., NAFL supports
the `axiom of closed ortho-curvature' stated in \cite{klr}
as follows:
\begin{quote}
That, with the axiom of closed ortho-curvature,
there are no true non-Euclidean geometries (and
no spatial dimensions beyond three), but only pseudo-geometries
consisting of curves in Euclidean space.
\end{quote}
Ross\cite{klr} also states two other axioms conjecturing the
existence of non-Euclidean geometries and then concludes that
these are questions in ``physics or metaphysics and are
logically entirely separate from the status of geometry in logic
or mathematics or from our psychological powers of visual
imagination''. In NAFL, however, the truth of the axiom
of closed ortho-curvature is upheld as a matter of logic
and the other axioms rejected; the axiomatic nature of
NAFL truth means that if we cannot \emph{in principle}
visualize non-Euclidean geometries, then they do not
exist. In this sense, NAFL vindicates Kant's position
that Euclidean geometry must be unavoidably true, although Kant did
not rule out (as does NAFL) the logical existence
of non-Euclidean geometries~\cite{klr}.

In conclusion, we state an argument for why it is impossible
to visualize non-Euclidean geometries in principle. Our contention
is that `curvature' of a line at a point is a notion that can
exist if and only if the associated center and radius of curvature in
the associated Euclidean space exist. It is tempting to assume that one can
`draw' a curve and find it to be `really' an arc of a circle without ever
making use of a compass. But the moment we assert that the drawn
curve is `really' an arc of a circle, we are also
unavoidably asserting the Platonic existence of the center and
radius of the circle and of the associated Euclidean space.
In the BK~model of HG noted above, one can certainly draw
arbitrary curved lines inside $\gamma$; this immediately
entails the Platonic existence of the entire Euclidean plane outside
$\gamma$ and means that circles of arbitrarily large radius can take
the place of $\gamma$ in the BK~model; as noted earlier,
this amounts to an explicit validation of
Euclid's fifth postulate. Similarly, in elliptic~(Riemannian)
geometry associated with a sphere, geodesics~(great circles) are
taken to be `intrinsically' straght lines; however, the associated
Platonic existence of the center of the sphere and radii
of curvature of the great circles makes the truth of
Euclidean geometry inevitable. In short, `true'
non-Euclidean geometries cannot be visualized even in principle
because of the unavoidable associated Platonic truth of
Euclidean geometry; the `models' of non-Euclidean geometries are
unavoidably Euclidean objects in Euclidean space.

\section*{Dedication}
The author dedicates this research to his son R.~Anand and wife
R.~Jayanti.

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