yet somehow it seems like at least 50 % of their weight is in their heads . I ' m convinced the only reason mine ' s neck is so big is because its necessary to support the ridiculous head on a dog thats only 38 lbs fully grown . <| sep |> I mean , aside from the absolutely rubbish circlejerking over how amazing engineers are ? Let ' s go through it piece by piece . I is just a wankfest about engineers while completely misunderstanding the issue . EG Focusingon electrical and electronic engineering , as a key example , the engineer iswell acquainted with the art of approximation . An engineer is trained tobe aware of the frailty of each modeland its limits when it breaks downBut platonists are perfectly alright with this , they never said every piece of mathematics is applicable to every part of reality , it ' s perfectly fine to say that the actual mathematical model of some system is complicated but a less complicated one will get close results . Hell , there ' s actually rigorous mathematical backing for this idea often times . That entire section is just a self congratulatory circle jerk that doesn ' t even really understand the issue . As for II , Penrose is seen as nutty , so the only really relevant section isA Platonist will argue that mathematicalforms follow from a set ofaxioms , and thus exist independentlyof our knowledge of them . This situationis no different to our lack of foreknowledgeof a fractal pattern , beforeexercising its originating equation . What can we say of the axioms themselves ? I argue that they are alsomental abstractions , and an exampleis given in Section V to illustrate thateven the simple counting of objectshas its physical limits . Thus , axiomsbased on the assumption of simplecounting are not universally real . But this paragraph is utter rubbish . The platonist wouldn ' t say that , since the platonist doesn ' t think mathematics is reducible to a formal system . Moreover , nobody looks at the axioms the guy is trying to discredit and goes yeah , those should hold everywhere . On III , So far , we have argued that mathematicsis a merely mental abstractionthat serves useful purposesNo , the dude asserted it randomly without understanding what he ' s arguing against . So this is again more circlejerking rhetoric without any substance . Another example is the use ofMaxwell ' s equations for modeling integratedelectromagnetic devices andstructures . In modern devices , due tothe complexity of design , we no longerresort to analytical calculationsinstead , electromagnetic simulationprograms that use numerical methodsare now the standard approach . The point here is that when wecarry out engineering in differentcircumstances , the way we performmathematics changes . Often thereality is that when analytical methodsbecome too complex , we simplyresort to empirical models andsimulations . Is similarly rubbish , the platonist doesn ' t think all math reduces to analytic solutions , it ' s pure posturing without any relevance to the issue . The Platonist will point out thatthe inverse square law for gravitationis spectacularly accurate at predictingthe behavior of nearby planets anddistant stars across vast scalesNo they won ' t . Partially because of the reasons he goes on to mention . This is just absurd and should be embarrassing to anyone reading it , he ' s assuming platonists don ' t know the first thing about physics and so base their views on outdated physics . Anyone seriously pushing this article should be ashamed for this utter claptrap . Note that mathematics has lessersuccess in describing biological systems , and even less in describing economicand social systems . But thesesystems have come into being and arecontained within our physical universe . Could it be they are harder tomodel simply because they adapt andchange on human time scales , and sothe search for useful invariant propertiesis more challenging ? Could it bethat the inanimate universe itself is nodifferent , but happens to operate on atimescale so large that in our anthropocentrismwe see the illusion ofinvariance ? Or it could be that it ' s a very complicated system and so it ' s difficult for us to make much headway . This is actually the accepted explanation within the mathematical community , that it ' s hard to model biological systems due to complexity , not that it ' s impossible . But Abbott doesn ' t even consider this . It ' s pure rhetoric , no argument . Moving on to the subsections , A Would hold for all of science as well . But we have decent reason to be scientific realists . Thus this objection must be further clarified in order to apply to mathematics . Does Abbott notice this fault and attempt to rectify it ? Of course not . BI argue that this innatenessis illusory , as we have cherry pickeda successful coincidence from abackdrop of many more cases that arenot as fortuitous . Abbott mistakes argue for claim . Abbott claims this , not argues it . But the problem here is that Abbott is quite , quite wrong . He brought up 4 d space time , so let me do him one better . Quantum Mechanics and various forms of algebra . Let ' s fucking listen to Peter Woit , shall we ? Pauli was scornful about this new , unphysical , mathematical formalism of matrices ~ woitwordpress ? p 191 Like holy shit man , those are the two most important physical theories ever and the math preceded them each time . Abbott is either ignorant of the history or being dishonest . C Taking into account the entire humanexperience , the number of questionsthat are tractable with scienceand mathematics are only a smallfraction of all the possible questionswe can askWho gives a shit ? Did any platonist say that math can answer any problem ? Next . Gdel ' s theorem also setlimits on how much we can actuallyprove . No it fucking doesn ' t you rank amateur . This only is true if you assume formalism from the get go . DA possibility is that the quest forsurvival has selected those who areable to follow chains of reasoning tounderstand local reality . This impliesthat the intellectual apparatus we useis in some way already appropriate . Hamming points out that , to someextent , we know that we are betteradapted to analyzing the world at ourhuman scale , given that we appear tohave the greatest difficulties in reasoningabout the very small scale andthe very large scale aspects of ouruniverse . Not an argument , an idle musing . So completely irrelevant to this discussion unless actual arguments are present . EThere is a fifth point we might addto Hamming ' s four propositions , andthat is that all physical laws and mathematicalexpressions of those lawsare a compression or compact representation . They are necessarily compresseddue to the limitations of thehuman mind . Therefore , they arecompressed in a manner suited tothe human intellect . The real world isinherently noisy and has a stochasticcomponent , so physical models areidealizations with the rough edgesremoved . Who gives a shit ? Again , no platonist said that physical reality was described by simple math . Abbott is just making up strawmen and knocking them down . FA sixth point we can add toHamming ' s list is that Wigner ' s senseof '' magic '' can be exorcised if we seethat the class of successful mathematicalmodels is preselected . Considerthe millions of failed models in theminds of researchers , over the ages , which never made it on paper becausethey were wrong . We tend to publishthe ones that have survived some levelof experimental vindication . Thus , this Darwinian selection process resultsin the illusion of automatic successour successful models are merelyselected out from many more failedonesAgain , applies to all of science , so needs to be further restricted unless we want to kill science as a guide to truth as well . But even past this , the claim is simply wrong , it doesn ' t kill the sense of magic . IVMathematical Platonists often pointout that a hypothetical alien civilizationwill most likely discover the number and put it to good use in theiralien mathematics . This is used to arguethat has its own Platonic existence , given that it is '' out there '' forany alien to independently discover . I mean , not really . It ' s a poor argument since the it ' s useful objection works here too . No sophisticated platonist uses this argument except against people too stupid to even think that far . So again , Abbott isn ' t bringing arguments , he ' s bringing rhetoric . VA deceptively simple example toillustrate the limitations in the correspondencebetween the ideal mathematicalworld and reality is to dissectthe idea of simple counting . Imaginecounting a sequence of , say , bananas . When does one banana end and thenext banana begin ? We think weknow visually , but to formally defineit requires an arbitrary decision ofwhat minimum density of bananamolecules we must detect to say wehave no banana . These are epistemological limitations at best , not ontological , so , again , either just rhetoric , or misunderstanding the issue . VIX are just rhetoric with no actual arguments presented .