William Tait - Academia.edu (original) (raw)

Papers by William Tait

Synthese, 1986

An arithmetical proposition A, for example, is about a certain structure, the system of natural n... more An arithmetical proposition A, for example, is about a certain structure, the system of natural numbers. It refers to numbers and relations among them. If it is true, it is so in virtue of a certain fact about this structure. And this fact may obtain even if we do not or (e.g., because of its relative complexity) cannot know that it does. This is a typical expression of what has come to be called the Platonist (or platonist or realist ) point of view toward mathematics. On the other hand, we learn mathematics by learning how to do thingsfor example, to count, compute, solve equations, and more generally, to prove. Moreover, we learn that the ultimate warrant for a mathematical proposition is a proof of it. In other words, we are justified in asserting A-and therefore, in any ordinary sense, the truth of A-precisely when we have a proof of it. Thus, we seem to have two criteria for the truth of A: it is true if (indeed, if and only if) it holds in the system of numbers, and it is true if we can prove it. But what has what we have learned or agreed to count as a proof got to do with what obtains in the system of numbers? I shall call this the Truth/Proof problem. It underlies many contemporary attacks on Platonism. The argument against Platonism begins with the observation that the first criterion, holding in the system of numbers, is inapplicable because we have no direct apprehension of this structure. Sometimes this argument is augmented by the thesis that 'apprehension of' would involve causal interaction with the elements of the structure and, since numbers are 'abstract' (i.e., not in spacetime), no such interaction is possible. In any case, the argument continues: it follows that a proof cannot be a warrant-or even incomplete evidence-for holding in the structure. For no kind of evidence is available that the canons of proof apply to the structure. Thus, if proof

Gödel's Reformulation of Gentzen's First Consistency Proof For Arithmetic: The No-Counterexample Interpretation

Bulletin of Symbolic Logic, 2005

The last section of “Lecture at Zilsel's” [9, §4] contains an interesting but quite condensed... more The last section of “Lecture at Zilsel's” [9, §4] contains an interesting but quite condensed discussion of Gentzen's first version of his consistency proof for PA [8], reformulating it as what has come to be called the no-counterexample interpretation. I will describe Gentzen's result (in game-theoretic terms), fill in the details (with some corrections) of Gödel's reformulation, and discuss the relation between the two proofs.

Advanced Materials, Jun 1, 2017

In article number 1700828, Dianyun Zhang, Luyi Sun, and co-workers develop a series of moisture-r... more In article number 1700828, Dianyun Zhang, Luyi Sun, and co-workers develop a series of moisture-responsive wrinkled-surface devices made using a poly(vinyl alcohol)-poly(dimethylsiloxane) bilayer system. These devices demonstrate reversible, temporary, or permanent wrinkles as moisturized, providing a route to tune the optical properties of a wrinkled surface for various applications, such as encryption devices, anticounterfeiting tabs, and water indicators.

RSC Advances, 2017

Conductive composite hydrogels based on sodium carboxymethyl cellulose (CMC) and polyaniline (PAn... more Conductive composite hydrogels based on sodium carboxymethyl cellulose (CMC) and polyaniline (PAn) were prepared via the semi-interpenetrating polymer network (semi-IPN) method, using glycerol diglycidyl ether (GDE) as the crosslinking agent. The structures of the resulting composite hydrogels were characterized by 13 C-NMR, FT-IR, and SEM. The composite hydrogels with various concentrations of CMC presented a similar swelling kinetic behaviour and a relatively high swelling ratio. With increasing CMC concentration, both the compressive fracture stress and modulus of the composite hydrogels improved gradually. Though the modulus of the composite hydrogels increased with more GDE added, the compressive fracture stress decreased when the crosslinking density was too high. The electrical conductivity of the composite hydrogels increased first and then decreased slightly with increasing dosages of CMC and GDE. Doping with aromatic sulfonate is a good approach to improve the conductivity of the composite hydrogels and the conductivity reached 6.31 Â 10 À3 S cm À1 after doping with sodium benzene sulfonate (BSNa).

Journal of Hazardous Materials, 2019

A water-based silsesquioxane (POSS)-containing polymer, POSS-PAA, was synthesized by using octavi... more A water-based silsesquioxane (POSS)-containing polymer, POSS-PAA, was synthesized by using octavinyl-POSS (V-POSS) and acrylic acid (AA) via interfacial polymerization. The TEM of POSS-PAA showed that the polymer formed a core-shell structure in aqueous solution and was well-dispersed. The star-shaped POSS-PAA and linear PAA were both tanned with 3.5% chromium tanning agent, and leather hide was tanned with 3.5% chromium tanning agent as a control. The results showed that the shrinkage temperature of wet-blue leather treated by POSS-PAA was increased by 3.5 ºC than that of the control. The thickening rate of the POSS-PAA treated wet-blue leather samples was increased by 21% and 96% than the linear PAA-treated leather and the control leather, respectively. The EDS results suggested that the POSS-PAA pretreated leather had a higher chromium content than the others, and the chromium distribution from the leather flesh side to the grain side was uniform. Moreover, the Cr2O3 content in both the POSS-PAA and the PAA pre-treated tanning waste water was reduced by about 50%, compared to the control waste. The COD and BOD of the POSS-PAA pre-tanning waste were decreased compared to the others. Therefore, POSS-PAA appeared to be promising for promoting the development of cleaner leather production.

Team Leading. Supervising: Principles and Practice of Supervision. The Choice Series #14. A Self Learning Opportunity

Journal of Materials Science, Oct 3, 2017

In this work, high purity amorphous silica derived from rice husk (RH) biomass was used to prepar... more In this work, high purity amorphous silica derived from rice husk (RH) biomass was used to prepare green phosphor (Zn 2 SiO 4 :Mn 2?). Based on the solid phase reaction under high temperature, the optimum doping concentration and reaction temperature were identified. The overall performance in terms of photoluminescence intensity and quantum yield of the RH-derived phosphor was superior to the one derived from commercial silica and close to the one made from silicic acid. The results showed that the phosphor derived from RH silica could serve as an alternative to commercial phosphor because of its decent properties and inexpensive green resource.

Tailoring Multistimuli Responsive Micropatterns Activated by Various Mechanical Modes

Advanced Functional Materials, Feb 24, 2021

A versatile wrinkle‐based micropatterned system with a film‐substrate structure is proposed, whic... more A versatile wrinkle‐based micropatterned system with a film‐substrate structure is proposed, which contains a hydrophilic film of polyvinyl alcohol or its composite with laponite and a hydrophobic substrate of polydimethylsiloxane or its composite with carbon black. The wrinkled system features high design flexibility and multistimuli responsiveness, which can be activated by various mechanical methods, including vertical press or scratch, gentle stretch‐and‐release, bend, or analogous magneto‐mechanical and electro‐mechanical modes. The resultant wrinkles possess 1) instantaneous and reversible strain/moisture/light responsive optical modulation; 2) tunable dynamics for the aforementioned strain/moisture/light response;3) tailorable amplitude/wavelength; 4) unique surface morphologies from the coupling of wrinkles and cracks; 5) excellent reversibility and durability. A variety of applications are demonstrated based on this system, including 1) a moisture erasable highly sensitive pressure responsive device and pattern replicator with a high fidelity; 2) a moisture erasable scratch/magneto‐mechanical re‐writable tablet; 3) an electro‐mechanical controllable smart window with an ultra‐sensitive strain responsive transmittance modulation and a low operating voltage; 4) various types of strain responsive, moisture erasable, and laser writable information recording/encryption devices. This work provides new routes for designing innovative wrinkled systems triggered by diverse mechanical fashions and can decode multiple environmental stimuli into optical signals for widespread application.

Advanced Materials, Apr 21, 2017

One-Pot Structure Direction of Large-Pore Co-Continuous Carbon Monoliths from Ultralarge Linear Diblock Copolymers

Chemistry of Materials, Oct 1, 2021

Part of the Chemical Engineering Commons, Environmental Engineering Commons, and the Materials Sc... more Part of the Chemical Engineering Commons, Environmental Engineering Commons, and the Materials Science and Engineering Commons Recommended Citation Tait, William, "Stretchable methyl ammonium lead iodide perovskite solar cells for photovoltaic and piezoelectric energy harvesting" (2018). University Scholar Projects. 40.

Journal of Leather Science and Engineering, Jan 15, 2021

Rice Husks (RHs) are one of the most abundant sources of biomass in the world due to rice consump... more Rice Husks (RHs) are one of the most abundant sources of biomass in the world due to rice consumption. Lignocellulose and silica are two of the main components of RHs, which allow RHs to be applied in different areas. Lignocellulose can be partially dissolved in 1-butyl-3-methylimidazolium chloride (BMIMCl), which is a simple way of competing with the traditional extraction methods that suffer from high chemical consumption. A lignocellulose freeze gel is obtained via a cyclic liquid nitrogen freeze-thaw (NFT) process. Multi-functional self-assembled lignocellulose aerogel is obtained after CO 2 supercritical drying. Based on the aerogel's special properties, two routes are developed for practical applications. On one hand, the aerogel is coated to exhibit a superhydrophobic property that can be applied as an absorbent for oil spills. On the other hand, a carbon aerogel is synthesized via a pyrolysis process, resulting in a porous amorphous carbon. The residue after partially dissolving lignocellulose in BMIMCl is further calcined to obtain amorphous silica nanoparticles, achieving a comprehensive application of RHs.

The Promise of Soft‐Matter‐Enabled Quantum Materials (Adv. Mater. 5/2023)

Advanced Materials, Feb 1, 2023

Taking Decisions. Supervising: Principles and Practice of Supervision. The Choice Series #20. A Self Learning Opportunity

The Promise of Soft‐Matter‐Enabled Quantum Materials

Advanced Materials, Dec 9, 2022

Triblock Terpolymer Thin Film Nanocomposites Enabling Two-Color Optical Super-Resolution Microscopy

Gentzen's Centenary, 2015

The story of Gentzen's original consistency proof for first-order number theory (Gentzen 1974), 1... more The story of Gentzen's original consistency proof for first-order number theory (Gentzen 1974), 1 as told by Paul Bernays (Gentzen 1974), (Bernays 1970), (Gödel 2003, Letter 69, pp. 76-79), is now familiar: Gentzen sent it off to Mathematische Annalen in August of 1935 and then withdrew it in December after receiving criticism and, in particular, the criticism that the proof used the Fan Theorem, a criticism that, as the references just cited seem to indicate, Bernays endorsed or initiated at the time but later rejected. That particular criticism is transparently false, but the argument of the paper remains nevertheless invalid from a constructive standpoint. In a letter to Bernays dated November 4, 1935, Gentzen protested this evaluation; but then, in another letter to him dated December 11, 1935, he admits that the "critical inference in my consistency proof is defective." The defect in question involves the application of proof by induction to certain trees, the 'reduction trees' for sequents (see below and §1), of which it is only given that they are well-founded. No doubt because of his desire to reason 'finitistically', Gentzen nowhere in his paper explicitly speaks of reduction trees, only of reduction rules that would generate such trees; but the requirement of wellfoundedness, that every path taken in accordance with the rule terminates, of course makes implicit reference to the tree. Gentzen attempted to avoid the induction; but as he ultimately conceded, the attempt was unsatisfactory. * My understanding of the philosophical background of Gentzen's work on consistency was enhanced by reading the unpublished manuscript " On the Intuitionistic Background of Gentzen's 1936 Consistency Proof and Its Philosophical Aspects" by Yuta Takahashi.

Materials Horizons, 2020

A 3D tubular structure fabricated from the relaxation of pre-stretched 2D precursor is used to de... more A 3D tubular structure fabricated from the relaxation of pre-stretched 2D precursor is used to develop various 3D stretchable electronics.

Annals of Vascular Surgery, 1991

Three cases of true aneurysms of the profunda femoris artery are reported along with a review of ... more Three cases of true aneurysms of the profunda femoris artery are reported along with a review of 17 other cases in the literature. These aneurysms are rare and commonly present with rapid enlargement or rupture (9/20), the risk of rupture being higher than those affecting the femoral or popliteal arteries. All patients underwent successful surgical treatment except for one who required amputation. The diagnosis of an aneurysm of the profunda femoris artery must be considered in all patients with a pulsatile swelling in the groin. Surgical treatment is mandatory, and it carries a low mortality as well as a low risk of amputation.

Intensional interpretations of functionals of finite type I

Journal of Symbolic Logic, 1967

T0 will denote Gödel's theory T[3] of functionals of finite type (f.t.) with intuitionistic q... more T0 will denote Gödel's theory T[3] of functionals of finite type (f.t.) with intuitionistic quantification over each f.t. added. T1 will denote T0 together with definition by bar recursion of type o, the axiom schema of bar induction, and the schemaof choice. Precise descriptions of these systems are given below in §4. The main results of this paper are interpretations of T0 in intuitionistic arithmetic U0 and of T1 in intuitionistic analysis is U1. U1 is U0 with quantification over functionals of type (0,0) and the axiom schemata AC00 and of bar induction.